An Uncertainty Augmentation Approach to Varying Analysis

(1)

An Uncertainty Augmentation Approach to Varying Analysis

34th CDC

Anders Helmersson

Department of Electrical Engineering Linkoping University

S-581 83 Linkoping, Sweden tel: +46 13 281622 fax: +46 13 282622 email:

andersh@isy.liu.se

February 27, 1995

Abstract

We present a concept for analysis and synthesis with varying uncertainties. We assume that the uncertainties are unknown by bounded and with bounded rate of variation. The approach taken here is to augment the uncertainty block with its derivative. This can be achieved using appropriate dynamic multipliers. The approach ts nicely in the framework and

^D

-

^K

-iteration-like methods can be used for synthesis.

Keywords:

Structured singular values, parametric dependent systems, time-varying systems.

1 Introduction

Systematic theories and tools for analysis and design of linear systems with uncertainties have been developed during the last decade and are still being rened and extended. With the advent of the two-Riccati-equation method for solving the

^H¹

control problem in 1988 2], the tools became more easy to use and the spread to new applications took a stride. Doyle 1] introduced the structured uncertainties and the

-formalism, where linear uncertainties can be of two types: real and complex. Real uncertainties are parametric, while complex uncertainties include linear dynamic e ects.

The

-analysis involves scaling matrices with a structure corresponding to the structure of the uncertainties. The scaling matrices are varied in order to optimize a criterion: either a matrix inequality or the maximum singular value.

The minimization of the maximum singular value can be reformulated into a

This work was supported by the Swedish National Board for Industrial and Technical

Development (NUTEK), which is gratefully acknowledged

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linear matrix inequality (LMI). If the uncertainties are constants the scaling matrices can be frequency dependent. If the uncertainties are varying without any bound on the rate of change the scaling matrices are constants. Recently methods for bridging the gap between constant and fast-changing uncertainties have been presented 6, 10, 11, 12].

In this paper an approach based on uncertainty augmentation is presented.

The main idea behind this approach is to introduce an uncertainty block that in addition to the uncertainty itself contains its derivative. With appropriate left and right dynamic multipliers the original uncertainty block can be recovered.

Using these multipliers we can reformulate the original problem into a standard

one.

The paper is organized as follows. In section 2 we review the

analysis concepts and linear fractional transformations (LFTs). Section 3 presents an approach for handling time varying uncertainties when these have bounded rate of variation. An example illustrating the method is given in section 4. A discussion on uncertainty structures and their classication follows in section 5.

How to include the approach in controller synthesis is discussed in section 6.

The conclusions are given in section 7.

1.1 Notations

The notations used are fairly standard. We use

^I

n to denote a unit matrix of dimension

ⁿ ⁿ

^X

denotes the complex conjugate transpose of

^X

^X^>

(

) 0 a hermitian (

^X

=

^X

) positive denite (semidenite) matrix

^X^;

= (

^X

)

^;1

ker

^X

denotes the null space of

^X

and range

^X

its range

^X^?

denotes a matrix such that ker

^X^?

= range

^X

and

^X^?^X^? ^>

0 note that

^X^?

only exists if

^X

has linearly dependent rows and that

^X^?

is not unique but in this paper any choice is acceptable

^X^y

is the Moore-Penrose pseudo inverse of

^X

diag

^X¹^X²

] a block-diagonal matrix composed of

^X¹

and

^X²

rank

^X

denotes the rank of the matrix

^X

herm

^X

=

¹²

(

^X

+

^X

)

^S

(

^:^:

) denotes the Redhe er star product

AB

denotes the Kronecker product of the matrices

^A

and

^B

(

^X

) the maximal singular value of

^X

and

^kGk¹

is the

^H¹

norm of the linear system

^G

.

2 -analysis and LFTs

This section gives a short review on structured singular values and linear fractional transformations (LFTs), see also e.g. 3].

2.1 Denitions

The denition of

depends upon the underlying block structure of the uncertainties , which could be either real or complex, see gure 1. See also

15, 16]. For notational convenience we assume that all uncertainty blocks are square. This can be done without loss of generality by adding dummy inputs or outputs.

Given a matrix

^M ²^C

ⁿ

ⁿ and two non-negative integers

^f

R , and

^f

C , with

f

=

^f

R +

^f

C

ⁿ

the block structure is an

^f

-tuple of pairs of positive integers

N K

= (

ⁿ¹^k¹

)

^:^:^:

(

ⁿ

f

^R^k

f

^R

)

(

ⁿ

f

^R+1^k

f

^R+1

)

^:^:^:

(

ⁿ

f

^R+

f

^C^k

f

^R+

f

^C

)]

(1)

(3)

M

-

Figure 1: System with uncertainties.

where

^P

^f _i

⁼¹ⁿ

i

^k

i =

ⁿ

for dimensional compatibility. The block repetition structure is dened by

^N

and the basic blocks by

^K

. The set of allowable perturba- tions is dened by a set of block diagonal matrices

^X ²^C

ⁿ

ⁿ dened by

X

=

^f

= diag

^I

n

¹

^R

¹^:^:^:^I

n

^f^R

_Rf

^R

I

n

^f^R⁺¹

_Cf

^R+1^:^:^:^I

n

^f^R^+f^C

_Cf

^R+

_f

^C

] :

_Ri

²^R

^k

ⁱ

^k

ⁱ

_Cf

^R+

_i

²^C

^k

^f^R⁺ⁱ

^k

^f^R⁺ⁱ^g

(2) where

denotes the Kronecker product:

AB

=

2

6

4 a

11 B a

12

B ::: a

1

n

^B

... ... ...

a

m

¹^B ^a

m

²^B ^:^:^: ^a

mn

^B

3

7

5 :

The Kronecker product has the following property, which is straight-forward to show,

(

^A^B

)(

^C^D

) = (

^AC

)

(

^BD

)

^:

(3) Using this equation we can now show the following commutative equation

(

^I

n

)(

^D^I

k ) = (

^I

n

^D

)

(

^I

k ) = (

^DI

n )

(

^I

k ) = (

^D^I

k )(

^I

n

) where

²^C

^k

^k and

^D²^C

ⁿ

ⁿ . That is,

^I

n

and

^D^I

k commute.

The uncertainty structure used here is slightly more general than in e.g.

15, 16], since also repeated full blocks are allowed both for real and complex uncertainties.

Assuming the uncertainty structure

^N ^K

, the structured singular value

of a matrix

^M²^C

ⁿ

ⁿ is dened by

=

min

2X

f

() : det(

^I^;

^M

) = 0

^g

;1

(4)

and if no

²^X

satises det(

^I^;

^M

) = 0 then

(

^M

) = 0.

2.2 Upper Bounds

Generally the structured singular value cannot be exactly computed, and instead

we have to resort to upper and lower bounds, which are usually sucient for

(4)

most practical applications. A tutorial review of the complex structured singular value is given in 9].

An upper bound can be determined using convex methods, either involving minimization of singular values with respect to a scaling matrix or by solving a linear matrix inequality (LMI) problem. The upper bound is conservative in the general case, but can be improved by branch and bound schemes.

A lower bound can be found by maximizing the real eigenvalue of a scaled matrix. This bound is nonconservative in the sense that if the true global maximum is found it is equal to

. However, since the problem is not convex, we cannot guarantee that we nd the global maximum.

We will here focus on the computation of the upper bound, which we here denote

, in order to distinguish it from the true

function. The upper bound

can be computed as a convex optimization problem. For complex uncertainties it is dened by

(

^M

) = inf _D

2D

(

^DMD^;1

)

(

^M

) (5) where

^D

is the set of block diagonal Hermitian matrices that commute with

^X

, that is

D

=

^f

0

^<^D

=

^D²^C

ⁿ

ⁿ :

^D

=

^D⁸

²^Xg:

(6) This problem is equivalent to an LMI problem

(

^M

) = inf _>

P

^2D0

f

:

^M^P^M^<²^Pg:

(7) Real uncertainties can be included in the LMI problem for computing the upper bound (see e.g. 4, 15, 16]). We dene

(

^M

) = inf _>

P G

^2D^2G0

f

:

^M^P^M

+

^j

(

^GM^;^M^G

)

^<²^P^g

(8) where

G

=

^fG

=

^G²^C

ⁿ

ⁿ :

^G

=

^G⁸

²^X^g:

(9) Every

^G²^G

is block diagonal with zero blocks for complex uncertainties. If we let

^G

=

^f

0

^g

in (8) we recover the complex upper bound (7).

We can reformulated (8) as a positive real property

(

^M

) = inf _>

W

^2W0

: herm

^;

(

^I

+

¹^M

)

^W

(

^I^;

¹^M

)

^>

0 (10) where herm

^X

=

¹²

(

^X

+

^X

) and

^W

=

^fW

=

^P

+

^jG

:

^P ²^D^G²^Gg

. Note that herm

^W ^>

0 always and that

^W

=

^W

for complex uncertainties. Another equivalent reformulation of (8) is

(

^M

) = inf _>

D G

^2D^2G0 n

:

^;

¹^DMD^;1^;^jG

(

^I

+

^G²

)

^;¹²^<

1

^o^:

(11)

(5)

2.3 Linear Fractional Transformations (LFTs)

Suppose

^M

is a complex matrix partitioned as

M

=

M

11 M

12

M

21 M

22

2C

(

p

¹⁺

p

²⁾⁽

m

¹⁺

m

²⁾

(12) and let u

²^C

m

¹

p

¹

and l

²^C

m

²

p

²

. The upper and lower linear fractional transformations (LFTs) are dened by

F

u (

^M

u ) =

^M²²

+

^M²¹

u (

^I^;^M¹¹

u )

^;1^M¹²

(13) and

F

l (

^M

l ) =

^M¹¹

+

^M¹²

l (

^I^;^M²²

l )

^;1^M²¹

(14) respectively. Clearly, the existence of the LFTs depends on the invertibility of

I;M

11

u and

^I^;^M²²

l respectively.

The Redhe er star product 13] is a generalization of the LFTs. Assume that

^Q

is partitioned similarly to

^M

. Then the star product is dened by

S

(

^Q^M

) =

F

l (

^Q^M¹¹

)

^Q¹²

(

^I^;^M¹¹^Q²²

)

^;1^M¹²

M

21

(

^I^;^Q²²^M¹¹

)

^;1^Q²¹ ^F

u (

^M^Q²²

)

:

(15) Note that the denition above is dependent on the partitioning of the matrices

^Q

and

^M

. The LFTs can be dened by the

^S

notation, as

F

u (

^M

u ) =

^S

( u

^M

) and

^F

l (

^M

l ) =

^S

(

^M

l )

^:

The star product is associative, that is

^S

(

^A^S

(

^B^C

)) =

^S

(

^S

(

^A^B

)

^C

)

^:

2.4 Frequency transformation

The bilinear transformation between the

^z

-domain and the

^s

-domain is

^s

= _z ^z

^;1⁺¹

, which is given by

^F

u (

^N^z^;1^I

) where

N

=

I

p

2

^I

;

p

2

^I ^;I

:

Using this transformation we can map the continuous time problem to a static

problem 3].

2.5 on Linear Systems

If

^G

(

^s

) =

^D

+

^C

(

^sI^;^A

)

^;1^B

is a stable linear system, we dene

kGk

= max _!

2R

f

(

^G

(

^j!

))

^g^:

(16) Note that this notation is somewhat misleading since

^k:k

is not a norm.

Using the frequency transformation

^s

= ^z _z

⁺¹^;1

together with LFTs we can recast the

^k:k

function into a pure

problem. We can then obtain the following LMI test for giving an upper bound on

^kGk

. If there exist

^>

0,

^P²^D

, ;

²^jG

and

^X ^>

0, such that

A

T

X

+

^X^A ^X^B

B

T

X

0 +

C D

0

^I

T

P

; ^T

;

^;²^P

C D

0

^I

<

0 then

^kGk

.

(6)

3 Time Varying Uncertainties

3.1 A Unied Approach

We will here adopt a unied approach for including time-varying uncertainties in the

formalism. We have previously based the computation of the upper

bound on two commutative sets

^D

and

^G

. In the case of time-invariant (constant) uncertainties, either parametric (real) or dynamic (complex), we may have multipliers

^D

and

^G

that are frequency dependent (dynamic). In the case of time-varying uncertainties, including nonlinear elements with bounded

^L²

gain and parametric varying parameters,

^D

and

^G

are generally restricted to constants (with respect to time or frequency).

Both these structures, constant and nonlinear, are extremes. A \constant"

parameter in practice is usually not constant but slowly varying. A dynamic nonlinear uncertainty is normally too conservative. Thus, it would be of great importance to include other structures in between these two extremes. We will here look at uncertainties that are varying with bounds on the rate of change.

The approach presented allows us to include these uncertainties in the standard

framework.

We denote uncertainties that have bounds on

_

^:^:^:

⁽

^m

⁾

, but not on

⁽

^m

⁺¹⁾

or higher derivatives, to belong to the class

^V

m . Any time-varying uncertainty (without bounds on _) belongs to

^V⁰

. We can include these time- varying structures into the framework presented here.

3.2 Uncertainty Block Augmentation

The main idea in this approach to handle slowly time-varying uncertainties is the concept of uncertainty block augmentation. This idea can be used also for constant uncertainties. An uncertainty block can be augmented by either adding copies of the original uncertainty or its derivative other choices may also be possible.

We have previously used the commutative property as an important tool for

nding the upper bound on

. The commutative property

^D

=

^D

can also be stated as

^D

^D^;1

= , since

^D

by denition is nonsingular.

Generally, we need no commutative set, but it is enough to nd an augmented uncertainty block ~ together with left and right (dynamic) multipliers,

^Y

(

^s

) and

Z

(

^s

) respectively, such that

Y

(

^s

)~

^Z

(

^s

) =

^:

If this is possible, then the system

^M

can be shown to be stable if

(~)

^<

1

^=kZMY^k

.

For a constant uncertainty

we can augment the uncertainty block by copies of itself: ~ =

^I

n . Then we can choose any dynamic multipliers

^Y

(

^s

) and

^Z

(

^s

) such that

^Y

(

^s

)

^Z

(

^s

) = 1. Thus,

^Y

(

^s

)~

^Z

(

^s

) =

.

Example 3.1 Let ~ = diag

],

^Y

(

^s

) =

¹² ¹⁺²¹⁺

^s _s

¹⁺²¹⁺

_s ^s

and

^Z

(

^s

) =

"

1+2

s

1+

s

1+

s

1+2

s

#

, then

^Y

(

^s

)

^Z

(

^s

) = 1 and consequently

^Y

(

^s

)~

^Z

(

^s

) =

.

²

(7)

For slowly time-varying uncertainties we include _ = _ddt into the augmented uncertainty block ~. We use

^s

to denote the di erential operator _ddt interchangeably with the Laplace argument.

Example 3.2 ^{Let ~ =}

_{_}

,

^Y

(

^s

) =

1 +

^as ^;a

and

^Z

(

^s

) = 1 1 +

^as

. Let

^y

= 1 1 +

^as^x

. Then

Y

(

^s

) ~

^Z

(

^s

)

^x

= (1 +

^as

)

^y^;^a^y

_

=

^y

+

^a^y

_ +

^a^y

_

^;^a^y

_

=

(

^y

+

^a^y

_ ) =

(1 +

^as

)

^y

=

^x:

Thus,

^Y

(

^s

)~

^Z

(

^s

) =

.

²

We can generalize this to the multivariable case, for which we provide the following lemma.

Lemma 3.1 Let =

^I

where

is a real or complex scalar. Assume that

^A

has all its eigenvalues with negative real part and let the system be initialized to zero state at

^t

=

^;1

. Then we have =

^Y

(

^s

)~

^Z

(

^s

), if either

(i) ~ =

_

,

^Y

(

^s

) =

^sI^;^A ^;I

and

^Z

(

^s

) = (

^sI^;^A

)

^;1

or (ii) ~ =

_

,

^Y

(

^s

) = (

^sI ^;^A

)

^;1

and

^Z

(

^s

) =

sI;A

I

.

Proof:

(i) Let

^y

= (

^sI^;^A

)

^;1^x

, then

Y

(

^s

)~

^Z

(

^s

)

^x

= (

^sI^;^A

)

^y^;^y

_

= _

^y

+

^y

_

^;^Ay^;^y

_

=

( _

^y^;^Ay

) =

(

^sI^;^A

)

^y

=

^x:

(ii)

Y

(

^s

) ~

^Z

(

^s

)

^x

= (

^sI^;^A

)

^;1

(

^sI^;^A

)

^x

+ _

^x

)

= (

^sI^;^A

)

^;1

(

^x

_

^;^Ax

+ _

^x

)

= (

^sI^;^A

)

^;1

(

^sI^;^A

)(

^x

) =

^x:

2

We can now combine the two results in lemma 3.1 into the following more general lemma.

Lemma 3.2 Let =

^I

where

is a real or complex scalar. Assume that

A

1

and

^A²

have all their eigenvalues with negative real part and let the system

(8)

be initialized to zero state at

^t

=

^;1

. Then with ~ = diag

_] we have

=

^Y

(

^s

)~

^Z

(

^s

) if

Y

(

^s

) =

^Y⁰

(

^s

)

^Y¹

(

^s

)

=

(

^sI^;^A¹

)

^;1

(

^sI^;^A²

) (

^sI^;^A¹

)

^;1

(17) and

Z

(

^s

) =

Z

0

(

^s

)

Z

1

(

^s

)

=

(

^sI^;^A²

)

^;1

(

^sI^;^A¹

) (

^sI^;^A²

)

^;1

(

^A¹^;^A²

)

:

(18)

Proof: We expand the -block by using lemma 3.1 (ii) and (i) in two consec- utive steps.

= (

^sI^;^A¹

)

^;1

_

sI;A

1

I

= (

^sI^;^A¹

)

^;1

sI;A

2

;I

_

(

^sI^;^A²

)

^;1

_

^sI^;^A¹

I

= (

^sI^;^A¹

)

^;1

(

^sI^;^A²

)(

^sI^;^A²

)

^;1

(

^sI^;^A¹

) + (

^sI^;^A¹

)

^;1^;

_(

^sI^;^A²

)

^;1

(

^sI^;^A¹

) + _

= (

^sI^;^A¹

)

^;1

(

^sI^;^A²

)(

^sI^;^A²

)

^;1

(

^sI^;^A¹

) + (

^sI^;^A¹

)

^;1

_(

^sI^;^A²

)

^;1

(

^A¹^;^A²

)

=

^Y

(

^s

)

0 0 _

Z

(

^s

)

^:

2

Remark 3.1 ~ If =

^I

k we can generalize the set of scaling matrices by letting

Y

(

^s

) =

^CY

(

^s

) and ~

^Z

(

^s

) =

^Z

(

^s

)

^B

where

^Y

and

^Z

are dened by (17) and (18) respectively and

^CB

=

^I

k .

Remark 3.2 We can also include !

and higher order derivatives in the uncertainty structure. The bounds on these higher order derivatives of

can be included easily in the same formalism, since the augmented block has the same properties as the original uncertainties. Hence, we can augment the _ -block with ! similarly to the augmentation of with _.

Remark 3.3 We have assumed that both

and _

are weighted equally in each block. Dierent weighting can be introduced by a scaling. Let ~ = diag

^=d

_ ] and scale the last block in

^Y

or

^Z

by

^d

.

Remark 3.4 The approach taken here is related to \the swapping lemma" 7]

used in adaptive control for stability analysis. The swapping lemma states in principle that the dierence

^D

(

^s

)

^;^D

(

^s

) is bounded. In our approach we explicitly solve the dierence and compensates for it exactly:

Z

0

(

^s

)

^;^Z⁰

(

^s

) =

^Z⁰

(

^s

)

^Y¹

(

^s

)_

^Z¹

(

^s

)

where we have used the fact that

^Z⁰

=

^Y⁰^;1

.

(9)

4 An Example

To illustrate the technique we give a simple example. The problem is described by

M

(

^s

) =

0

¹⁺⁴⁴⁺

^s _s

1+4

s

4+

s 0

=

1

0 0

²

1

2

2C

(19) for which

^kM^k

= 1 with time-invariant uncertainties. To obtain this we can use the dynamic scaling

D

(

^s

) = diag 1

4 +

^s

1 + 4

^s

]

since

kMk

=

^kDMD^;1^k¹

=

0 1 1 0

= 1

^:

Assuming dynamic uncertainties then

^kM^k

= 4 with a constant

^D

=

^I

. In order to make the system unstable using a nonlinear uncertainty already at a low gain

^> ¹⁴

, we need uncertainty elements that are able to transform energy from one frequency to another. If the uncertainty elements are slowly varying we may expect a lower value on

, which means that we can tolerate larger uncertainties while maintaining stability.

We now assume that the uncertainty

²²^V¹

is slowly time-varying (

¹²^V⁰

may change without bound on rate). We will use the following multipliers

Y

(

^s

) =

"

1 0 0

0

¹⁺⁴⁴⁺

_s ^s

^p⁴⁺¹⁵

^d= _s

⁴

#

and

^Z

(

^s

) =

2

6

4

1 0 0

¹⁺⁴⁴⁺

^s _s 0

^p¹⁺⁴⁶⁰

^d _s

3

7

5 :

that satises the structure given in lemma 3.2. Then

=

^Y

(

^s

) ~

^Z

(

^s

) =

^Y

(

^s

)

2

6

4

1

0 0

0

²

0 0 0 _

²^=d

3

7

5 Z

(

^s

)

where ~ is the augmented uncertainty. The scaled system now becomes

Z

(

^s

)

^M

(

^s

)

^Y

(

^s

) =

2

6

4

0 1

^p¹⁺⁴¹⁵

^d= _s

⁴

1 0 0

p

60

d

4+

s 0 0

3

7

5 :

The following upper bound is then obtained

kMk

^kZMY^k

0

B

@ 2

6

4

0 1

^p¹⁵²

^d

1 0 0

p

15

d

2

0 0

3

7

5 1

C

A

=

^q

1 +

¹⁵⁴^{d :}

We can now choose

^d

in order to cope with varying uncertainties. If we choose

d

to be small, the

value will be approximately 1. For instance, let

^d

= 0

^:

1

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10⁻² 10⁻¹ 10⁰ 10¹ 10² 0

0.2 0.4 0.6 0.8 1

d

1/mu

Varying mu analysis

Figure 2: Varying

analysis. The horizontal line through 1

⁼

= 0

^:

25 is the bound obtained with a constant scaling (no bound on the rate of

). The bell- shaped line starting close to 1 and then decreasing is the obtain with the scaling described in the example. The crosses mark improved bounds on 1

⁼

obtained using modied scalings.

then

1

^:

1726. Thus the system is stable if

^j¹^j^<

0

^:

8528,

^j²^j^<

0

^:

8528 and

j

_

²^j^<

0

^:

0853, since

(~) = max

^fj¹^j^j²^j^j

_

²^j=dg

.

If we choose

^d

to be large, the

value increases. For instance, with

^d

= 1 we obtain

2

^:

1794 and thus the system is stable if

^j¹^j^<

0

^:

4588,

^j²^j^<

0

^:

4588 and

^j

_

²^j

0

^:

4588. The bound can be improved further, especially for large values of

^d

by modifying

^Y

and

^Z

.

The result obtained from this analysis is given in gure 2. The two lines gives the bounds obtained from constant scaling and the augmented uncertainty technique. The bounds can be further improved by modifying the scalings (

^Y

and

^Z

). The crosses in the gure indicate such improved bounds.

5 Uncertainty Structures

In this section, we will discuss the various structures of uncertainties that can be included in the

framework. The denition of the uncertainty structure is closely related to its commuting properties. We will here make a more general classication of uncertainties than in e.g. 15, 16].

5.1 Classication

The uncertainties can be described by the four characteristics given in the table

below.

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1 eld (

^F

) complex (

^C

) real (

^R

)

dynamic parametric

2 structure full matrix repeated scalar

N K

without structure or full blocks 3 variability time-varying constant

V V

0

4 accessibility unknown shared

robustness gain scheduling

The table is arranged in such a way that more conservative results are obtained to the left of the table. Going to the right generally improves the bound, since it is less conservative. Also, more structure is needed.

Each of these four characteristics can be combined arbitrarily and each block in the uncertainty structure can be described by a four-tuple. However, not all combinations of uncertainty structures need to be meaningful or possible to interpret.

5.2 Complex and Real Uncertainties

The rst characteristic describes the if the uncertainty is complex or real. Com- plex uncertainties describe elements with dynamics and are used to model time delays and phase changes. This includes state-space realizations with time- varying coecients. Real numbers denote parametric uncertainties without any inherent dynamics.

5.3 Uncertainty Block Structure

The structure of the uncertainty block is described by the

^N ^K

where

^N

denes the repetition structure and

^K

the basic block structure. The basic block structure can be any (square) matrix with no inherent structure. The only matrix commuting with this structure is

^dI

where

^d

is a scalar.

The repeated block structure

^I

i = diag i

i

^:^:^:

i ]) has more structure. The corresponding scaling matrix is

^D^I

. Typically the basic block is a scalar and then the structure is called a repeated scalar block (

^I

) this structure has only one degree of freedom.

5.4 Time Varying Uncertainty

The time-varying property has two extremes: constant and time-varying without bound on rate. For constant uncertainties we can use dynamic scalings, while for time-varying uncertainties we are restricted to constant scalings.

In between these (not indicated in the table) we can included time-varying

uncertainties with bounds on _

and higher order derivatives. These can be han-

dled using the proposed method by using an augmented uncertainty structure

and related multipliers.

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5.5 Unknown and Shared Uncertainties

The uncertainty structure can either be unknown or shared. If we are dealing with unknown uncertainties the problem is a robust design problem. In this case we only know the structure of the uncertainty and that it is bounded.

In some applications the uncertainties are known or accessible to the controller. We are then dealing with a gain scheduling problem. This improves the performance compared to the robust (unknown) case since the controller can change its behavior according to the uncertainty.

6 Synthesis

The approach presented here can be combined with synthesis of controller for systems with time-varying uncertainties.

6.1 Robust Design

In the case of constant uncertainties so called

^D

-

^K

iterations are used. This includes two steps: rst nding a scaling system

^D

that minimizes the

value for a xed controller and then nding a controller for

^DMD^;1

using

^H¹

synthesis while keeping

^D

xed. These two steps are repeated until the

value converges or gets below the specied requirement. Even if there is no guarantee that the global optimum is found, the method works well in many applications.

The original

^D

-

^K

method was limited to complex uncertainties. Real uncertainties are a subset of complex ones and using

^D

-

^K

iterations in this case yield conservative results. The method can be improved to handle real uncertainties by generalized schemes 14].

By replacing

^D

and

^D^;1

with the multipliers

^Z

and

^Y

, the

^D

-

^K

iteration scheme can be extended to handle time-varying uncertainties as well.

6.2 Gain Scheduling Design

Gain scheduling controllers can be synthesized by methods based on

analysis

8, 5]. The uncertainty is in this case accessible to or shared by the controller.

This allows for improved performance compared to the robust case. Also, the problem of nding a gain scheduling controller is convex, which guarantees that the global solution is found. This is generally not the case for robust design.

The gain scheduling technique can be extended to time-varying uncertainties by the scaling the system by

^Z

and

^Y

. By applying the gain scheduling techniques mentioned above, we can hopefully improve the performance and reduced conservativeness.

7 Conclusions

In this paper an approach for

analysis with time-varying uncertainties with bounded rate of variation is presented.

The idea behind the approach is to augment the original uncertainty struc-

ture with its derivative. The paper gives a set of multipliers

^Y

and

^Z

that,

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when applied to the augmented uncertainty structure, recover the original uncertainty. Using this technique we can include bounds on the variation of the uncertainty while still keeping to the normal

framework.

For instance controller design can be made using

^D

-

^K

-like synthesis where the

^D

and

^D^;1

scalings are replaced by generalized multipliers

^Z

and

^Y

respectively.

References

1] J. C. Doyle. Analysis of feedback systems with structured uncertainties.

IEE Proceedings, 129, Part D(6):242{250, November 1982.

2] J. C. Doyle, K. Glover, P. Khargonekar, and B. A. Francis. State-space solutions to the standard

^H²

and

^H¹

control problems. IEEE Transactions on Automatic Control, 34(8):831{847, August 1989.

3] J. C. Doyle, A. Packard, and K. Zhou. Review of LFTs, LMIs, and

. In IEEE Proceedings of the 30th Conference on Decision and Control, volume 2, pages 1227{1232, Brighton, England, December 1991.

4] M. Fan, A. Tits, and J. C. Doyle. Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Transactions on Automatic Control, 36(1):25{38, January 1991.

5] A. Helmersson.

synthesis and LFT gain scheduling with mixed uncertainties. In European Control Conference. Submitted to ECC 1995.

6] U. J!onsson and A. Rantzer. System with uncertain parameters | time- variations with bounded derivatives. In Proceedings of the 33rd Conference on Decision and Control, volume 3, pages 3074{3079, Lake Buena Vista, Florida, December 1994.

7] K. Narendra and Annawamy A. Stable Adaptive Systems. Information and System Science Series. Prentice Hall, 1989.

8] A. Packard. Gain scheduling via linear fractional transformations. Systems

& Control Letters, 22(2):79{92, February 1994.

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10] A. Rantzer. Uncertain real parameters with bounded rate of variation. In K. J. $Astr!om, G. C. Goodwin, and P. R. Kumar, editors, Adaptive Control, Filtering and Signal Processing. Springer-Verlag. to appear.

11] A. Rantzer. Uncertainties with bounded rates of variation. In Proceedings of the American Control Conference, volume 1, pages 29{30, San Fransisco, California, 1993.

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13] R. Redhe er. Inequalities for a matrix Riccati equation. Journal of Math- ematics and Mechanics, 8(3), 1959.

14] P. Young. Controller design with mixed uncertainties. In Proceedings of the American Control Conference, volume 2, pages 2333{2337, Baltimore, Maryland, June 1994.

15] P. Young, M. Newlin, and J. Doyle.

analysis with real parametric uncertainties. In IEEE Proceedings of the 30th Conference on Decision and Control, volume 2, pages 1251{1256, Brighton, England, December 1991.

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An Uncertainty Augmentation Approach to Varying Analysis