An IQC-Based Stability Criterion for Systems with Slowly Varying Parameters

(1)

An IQC-Based Stability Criterion for Systems with Slowly Varying Parameters

Anders Helmerssonn

Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden

www:

http://www.control.isy.liu.se

email:

andersh@isy.liu.se

LiTH-ISY-R-1979 September 10, 1997

REGLERTEKNIK

AUTOMATIC CONTROL

LINKÖPING

Technical reports from the Automatic Control group in Linkoping are available as UNIX-compressed Postscript les by anonymous ftp at the address

130.236.20.24

(ftp.control.isy.liu.se)

.

(2)

An IQC-Based Stability Criterion for Systems with Slowly Varying Parameters

Anders Helmersson

Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden

www:

http://www.control.isy.liu .se

email:

andersh@isy.liu.se

Submitted to ACC 1998 September 10, 1997

Abstract

An integral quadratic constraints (IQC) is introduced for stability analysis of linear systems with slowly varying parameters. The parameters are assumed to be bounded and with bounded derivatives. Other types of uncertainties can be included in the problem. The new criterion yields signicantly less conservative bounds than previously proposed criteria.

Keywords:

Stability analysis, robustness, linear time-varying, integral quadratic constraints.

1 Introduction

Integral Quadratic Constraints (IQC) has recently been proposed by Megretski and Rantzer as a unied approach to robustness analysis 5].

We will here elaborate on IQCs for verifying stability of linear systems with slowly varying parameters. We assume that the parameters are bounded as well as are their derivatives. An IQC-based stability criterion for slowly-varying parameters (uncertainties) has been proposed by Jonsson and Rantzer 3, 2].

The IQC proposed in this paper exploits more of the structure of the problem than the previous criterion. Since higher exibility in the IQC multiplier can be allowed the new criterion gives less conservative bounds. This is illustrated by two examples.

The paper is organized as follows. Integral Quadratic Constraints are briey introduced in section 2. In section 3, the main result is elaborated including the swapping lemma and choice of multipliers. The new algorithm is applied to two examples in section 4. The conclusions are given in section 5.

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

Development (NUTEK), which is gratefully acknowledged

(3)

G

(

^s

)

6

-

+

i ^-

?

+

i f

v

e w

Figure 1: Basic feedback conguration.

1.1 Notation

For matrices

^A

denotes the complex conjugate transpose.

RH

1

denotes denotes stable real-rational transfer functions

^L2

denotes the Hilbert space of measurable functions

^R^!^Rⁿ

satisfying

kfk22

=

^Z ¹

;1

jf

(

^t

)

^j₂₂^dt^<¹

This is a subspace of

^L2^e

, whose members only need to be square integrable on

nite intervals.

2 Integral Quadratic Constraints

The integral quadratic constraints (IQCs) has been proposed for robustness analysis 5]. The IQC states a stability criterion for the interconnection of a stable system

^M²^RH¹

and a bounded causal operator , see gure 1.

(

v

=

^Mw

+

^f

w

=

^v

+

^e:

(1)

We say that the interconnection of

^M

and is well-posed if the map (

^v^w

)

^!

(

^e^f

) dened by (1) has a causal inverse on

^L2^e

. The interconnection is stable if, in addition, the inverse is bounded, that is if there exists a constant

^C

such that

Z

T

0

(

^jv

(

^t

)

^j²

+

^jw

(

^t

)

^j²

)

^dt^C

Z

T

0

(

^jf

(

^t

)

^j²

+

^je

(

^t

)

^j²

)

^dt

for any

^T

0 and for any solution of (1).

Depending on the particular application, various versions of IQCs are available. Two signals

^w² ^L2

0

¹

) and

^v ² ^L2

0

¹

) are said to satisfy the IQC dened by , if

Z

1

^v

^ (

^j!

)

^

w

(

^j!

)

(

^j!

)

^v

^ (

^j!

)

^

w

(

^j!

)

d!

0 (2)

where absolute integrability is assumed. Here ^

^v

(

^j!

) and ^

^v

(

^j!

) represent the harmonic spectrum of the signals

^v

and

^w

at the frequency

^!

. In principle

:

^jR ^!^C

can be any measurable Hermitian-valued function. In most appli-

cations, however, it is sucient to use rational functions that are bounded on

the imaginary axis.

(4)

A time-domain form of (2) is

Z

1

0

(

^x

(

^t

)

^v

(

^t

)

^w

(

^t

))

^dt

0 (3) where

is a quadratic form, and

^x

is dened by

_

x

(

^t

) =

^A^x

(

^t

) +

^B^v^v

(

^t

) +

^B^w^w

(

^t

)

^x

(0) = 0 where

^A

is a Hurwitz matrix.

The main theorem from 5] goes as follows

Theorem 1 (5]) ^Let

^G²^RH¹

^{and let} be a bounded causal operator. As- sume that:

i) for every

²

0 1], the interconnection of

^G

and

is well-posed

ii) for every

²

0 1], the IQC dened by is satised by

iii) there exists

^>

0 such that

G

(

^j!

)

I

(

^j!

)

G

(

^j!

)

I

;I 8!2R:

(4) Then the feedback interconnection of

^G

and is stable.

Note that if the upper left corner,

11

, of is positive semidenite then = 0 satises (2). If further the lower right corner,

22

, is negative semidenite then any convex combination of 's satisfying (2) also satises the IQC. Thus,

11

0 and

22

0 imply that

satises (2) for

²

0 1] if and only if does so. This simplies assumption ii).

This follows by the fact that

I

1

+ (1

^;

)

2

11

12

22

I

1

+ (1

^;

)

2

=

I1

11

12

₁₂

22

I1

+ (1

^;

)

I2

11

12

₁₂

22

I2

;

(1

^;

)

0

1^;

2

0 0 0

22

1^;

0

2

I1

11

12

₁₂

22

I1

+ (1

^;

)

I2

11

12

₁₂

22

I2

if

²

0 1] and

22

0. Also,

11

0 assures that the inequality is satised for = 0.

The search for multipliers, , can be carried out as a convex optimization problem by parametrizing

(

^j!

) =

^X

i x

i

ⁱ

(

^j!

)

where

^xⁱ

are positive real parameters and

ⁱ

is a set of basis multipliers. By

applying the Kalman-Yakubovich-Popov lemma 7, 8, 6], the search for

^xⁱ

, can

be implemented using linear matrix inequalities (LMIs).

(5)

3 Slowly Varying Uncertainties

We will here propose an IQC for verifying stability of linear system with slowly varying uncertainties. We dene slowly varying uncertainties such that their rate of change _ is norm-bounded by

^d

, that is

^k

_

^k^d

.

The main tool for this analysis is the swapping lemma that tells how a linear operator on state-space form commutes with a time-varying uncertainty.

3.1 Swapping Lemma

The following swapping lemma is essentially from 2].

Lemma 1 (Swapping Lemma) Suppose that ^ has the derivative _^ and

^u²

L2

. Further let

^T ²^RH¹

:

T

(

^s

) = ^

^D

+ ^

^C

(

^sI^;^A

^ )

^;¹^B

^

U

(

^s

) = ^

^C

(

^sI^;^A

^ )

^;¹

V

(

^s

) = (

^sI^;^A

^ )

^;¹^B

^ such that

^A

^

^B

^

C ^D

^

^ 0 0

=

^ 0 0

^

A ^B

^

C ^D

^

:

(5)

Then

^T

(

^s

) =

^T

(

^s

)

^u

+

^U

(

^s

) _^

^V

(

^s

)

^u:

(6)

Proof: ^Let

^s

⁼

^dt^d

. We note that

(

^sI^;^A

^ )^

^V

(

^s

)

^u

= _^

^V

(

^s

)

^u

+ ^

^Bu

= _^

^V

(

^s

)

^u

+

^B

^u:

(7) Let

^U

(

^s

) operate from the left on (7). After addition of ^

^Du

= ^

^D

^u

, we get

^

^D

+ ^

^C

^

^V

(

^s

)

^u

=

^U

(

^s

) _^

^V

(

^s

) + ^

^D

+

^U

(

^s

)

^B

^u

which is equivalent to (6).

²

Note that we can generalize the swapping lemma to also apply to a more general set of linear operators ^ for which we dene _^ by

d

dt

(^

^v

) = _^

^v

+ ^_

^v

for all

^v^v

_

²^L2

.

Also note that the block-diagonal structure of

^T

,

^V

and

^U

is implicitly dened by the commuting equation (5).

The set of uncertainties, , can be assumed to have a blockdiagonal structure, possibly with repeated sub-blocks:

=

^f

= diag

^Iⁿ¹

1^:^:^:^I^nf

^f

]

^g

(6)

where

denotes the Kronecker product. The uncertainties consist of

^f

diagonal blocks, which could be either dynamic (complex) or parametric (real). For parametric blocks also

ⁱ

=

ⁱ

applies. It is easy to show that for any

ⁱ ²

C

kiki

and

^Dⁱ²^Cⁿⁱⁿⁱ

the following commutative equation holds:

(

^Iⁿⁱ

ⁱ

)(

^Dⁱ^I^ki

) = (

^Dⁱ^I^ki

)(

^Iⁿⁱ

ⁱ

) (8) In the paper the structure of is implicit, and instead the structure is dened by commuting properties, such as (5) and (8).

For instance, if we assume that = diag

1^Iⁿ¹

2

] where

1

is a slowly varying parameter and

2

contains remaining uncertainties, a natural choice of

T

would be to use

^A

^

^B

^

C ^D

^

=

2

6

4

^

A ^B

^ 0

I

0 0 0

^I

0 0 0

^I

3

7

5

:

(9)

Then ^ =

1^I^k¹

, = diag

1^I^k¹

] = diag

1^Iⁿ¹+^k¹

2

], where

^k1

is the dimension of ^

^A

. In general ^ contains the slowly varying parameters in possibly (di!erently) repeated.

3.2 IQC Formulation

We will study a stability criterion for the system

^x

=

^Gx

where and _^ are norm-bounded. Consider the following IQC-like matrix inequality.

2

6

4

T

(

^j!

)

^G

(

^j!

) 0

V

(

^j!

)

^G

(

^j!

) 0

T

(

^j!

)

^U

(

^j!

)

0

^I

3

7

5

~

2

6

4

T

(

^j!

)

^G

(

^j!

) 0

V

(

^j!

)

^G

(

^j!

) 0

T

(

^j!

)

^U

(

^j!

)

0

^I

3

7

5

<

0 (10) where ~ = ~ is a static matrix. We can rewrite (10) as a proper IQC for the augmented system

^G

0 :

2

4

G

(

^j!

) 0

I

0

^I

3

5

(

^j!

)

2

4

G

(

^j!

) 0

I

0

^I

3

5

<

0

^!²^R^f1g

(11) where

=

11

12

22

=

2

6

4

T

0 0

V

0 0 0

^T ^U

0 0

^I

3

7

5

~

2

6

4

T

0 0

V

0 0 0

^T ^U

0 0

^I

3

7

5

:

(12)

The multiplier dened by (12) satises the IQC for the augmented uncertainty given by

"

_^

^V

#

. To show this we rst observe that the augmented uncertainty

(7)

is bounded since

^T ² ^RH¹

and, consequently, _^

^V

is bounded if _^ is such.

Using the swapping lemma, we obtain

w

=

2

6

4

T

0 0

V

0 0 0

^T ^U

0 0

^I

3

7

5 2

4

I

_^

^V

3

5

v

=

2

6

4 T

V

T

+

^U

_^

^V

_^

^V

3

7

5 v

=

2

6

4 T

V^T

_^

^V

3

7

5 v:

Thus (3) becomes

Z

1

0 ^w

(

^t

) ~

^w

(

^t

)

^dt

=

Z

1

0

w

T

(

^t

)

w

V

(

^t

)

2

6

4

I

0

^I

(

^t

) 0 0 _^(

^t

)

3

7

5

~

2

6

4

I

0

^I

(

^t

) 0 0 _^(

^t

)

3

7

5

w

T

(

^t

)

w

V

(

^t

)

dt

0 (13) where

^w^T

=

^T^v

and

^w^V

=

^V^v

are dened by

_^

x

(

^t

) = ^

^A^x

^ (

^t

) + ^

^Bv

(

^t

)

^x

^ (0) = 0 (14)

w

T

(

^t

)

w

V

(

^t

)

=

^C

^

^D

^

I

0 ^

x

(

^t

)

v

(

^t

)

:

Using (5) we have

"

(

^t

) 0 0 _^(

^t

)

#

^C

^

^D

^

I

0 =

^C

^

^D

^ 0 0 0

^I

2

6

4

^(

^t

) 0 0 (

^t

) _^(

^t

) 0

3

7

5 :

Consequently, (13) is equivalent to

Z

1

0

^x

^ (

^t

)

v

(

^t

)

2

6

4

I

0

^I

^(

^t

) 0 0 (

^t

) _^(

^t

) 0

3

7

5

~~

2

6

4

I

0

^I

^(

^t

) 0 0 (

^t

) _^(

^t

) 0

3

7

5

^x

^ (

^t

)

v

(

^t

)

dt

0 (15) for any

^v²^L2

and ^

^x

as dened by (14) and where

~~ =

2

4

~~

11

~~

12

~~

₁₂

~~

22

3

5

=

2

6

4

^

C ^D

^ 0 0 0

I

0 0 0 0 0 0 ^

^C ^D

^ 0 0 0 0 0

^I

3

7

5

~

11

~

12

~

₁₂

~

22

2

6

4

^

C ^D

^ 0 0 0

I

0 0 0 0 0 0 ^

^C ^D

^ 0 0 0 0 0

^I

3

7

5 :

(16)

If ~~

11

0 and ~~

22

0, then the inequality (15) is satised for any time-varying

convex combinations of ^'s that satisfy the same inequality. See the remark on

theorem 1.

(8)

We will now show that there exists such a multiplier, ~~, if

11

(

^j!

)

0 and

22

(

^j!

)

0 for all

^! ² ^R

. We assume that ^

^A

is Hurwitz and that the state-space representation of

^T

is minimal so that ( ^

^A^B

^ ) is controllable.

The positive semideniteness of

11

(

^j!

) is equivalent to

T

(

^j!

)

V

(

^j!

)

~

11

T

(

^j!

)

V

(

^j!

)

=

(

^j!I^;^A

^ )

^;¹^B

^

I

~~

11

(

^j!I^;^A

^ )

^;¹^B

^

I

0 for all

^! ² ^R

. Using the Kalman-Yakubovich-Popov lemma 7, 8, 6], this inequality holds if and only if there exists

^Y

=

^Y

such that

~~

11

+

Y^A

^ + ^

^A ^Y ^Y^B

^

B Y

0

^:

Thus, if

11

(

^j!

)

0 for all

^! ² ^R

we can modify ~~

11

to become positive semidenite by adding

~~

11^Y

=

Y^A

^ + ^

^A ^Y ^Y^B

^

B Y

0

:

This does not a!ect neither (11) nor (15), since

(

^j!I^;^A

^ )

^;¹^B

^

I

~~

11^Y

(

^j!I^;^A

^ )

^;¹^B

^

I

= 0

^:

Using the same sequence of arguments

22

(

^j!

) =

T

(

^j!

)

^U

(

^j!

)

0

^I

~

22

T

(

^j!

)

^U

(

^j!

)

0

^I

(17)

=

2

4

(

^j!^;^A

^ )

^;¹^B

^ (

^j!^;^A

^ )

^;¹

I

0

^I

3

5

~~

22

2

4

(

^j!^;^A

^ )

^;¹^B

^ (

^j!^;^A

^ )

^;¹

I

0

^I

3

5

0 holds for all

^!²^R

, if and only if there exists

^X

=

^X

such that

~~

22

+

2

4

X^A

^ + ^

^A ^X ^X^B

^

^X

^

B X

0 0

X

0 0

3

5

0

^:

(18)

Thus we can modify ~~

22

to become negative semidenite if and only if

22

(

^j!

)

0 for all

^!²^R

. Further, we can show that also (15) is una!ected by the same modication of ~~

22

, since

Z

1

0

^

^x

(

^t

)

v

(

^t

)

2

6

4

^(

^t

) 0 0 (

^t

) _^(

^t

) 0

3

7

5 2

4

X^A

^ + ^

^A ^X ^X^B

^

^X

^

B X

0 0

X

0 0

3

5 2

6

4

^(

^t

) 0 0 (

^t

) _^(

^t

) 0

3

7

5

^x

^ (

^t

)

v

(

^t

)

dt

=

Z

1

0

^

^x

(

^t

) ^(

^t

)

^X

( ^

^A

^(

^t

)^

^x

(

^t

) + ^

^B

(

^t

)

^v

(

^t

) + _^(

^t

)^

^x

(

^t

)) + ()

^dt

=

Z

1

0

^

^x

(

^t

) ^(

^t

)

^X

(^(

^t

) ^

^A^x

^ (

^t

) + ^ ^

^Bv

(

^t

) + _^(

^t

)^

^x

(

^t

)) + ()

^dt

=

Z

1

0

d

dt

(^

^x

(

^t

) ^(

^t

)

^X

^(

^t

)^

^x

(

^t

))

^dt

= lim

t!1

^

x

(

^t

) ^(

^t

)

^X

^(

^t

)^

^x

(

^t

)

^;^x

^ (0) ^(0)

^X

^(0)^

^x

(0) = 0

(9)

since ^

^x^x

_^

²^L2

and ^

^x

(0) = 0. Here we have used () to denote the transpose of the previous terms.

We have shown that if

11

(

^j!

)

0 and

22

(

^j!

)

0 for

^!²^R

, then we can

nd an equivalent IQC with such that

11

0 and ~~

22

0. Thus (13) and (15) both hold for any time-varying convex-cone combination such that

^(

^t

) =

^X

i

(

^t

) ^

ⁱ

(

^t

) (19a) _^(

^t

) =

^X

i

(

^t

) _^

ⁱ

(

^t

) (19b) for

ⁱ

(

^t

)

²

0 1] and

^Pⁱⁱ

(

^t

)

1. Note that (19) constrains

ⁱ

dynamically since ^ and _^ are related. Specically, it follows that

^Pⁱ

_

i

(

^t

) ^

ⁱ

(

^t

) = 0. For instance, if _^

ⁱ

= 0 then only time-invariant ^ are allowed.

If we have a multiplier ~ such that

2

6

4

I

0

^I

ⁱ

(

^t

) 0 0 ^

ⁱ

(

^t

)

3

7

5

~

2

6

4

I

0

^I

ⁱ

(

^t

) 0 0 ^

ⁱ

(

^t

)

3

7

5

0

⁸ⁱ^t

(20) that satises (11) and if

11

(

^j!

)

0 and

22

(

^j!

)

0 for all

^!²^R

, then the system is veried to be stable for any time-varying convex-cone combination of the

ⁱ

and _^

ⁱ

.

For instance, if =

^I

and _^ = _

^I

such that

^jj

1 and

^j^j

_

^<^d

, then we may use

~ =

2

6

4

P1

0 ;

1

0

^d²^P2

0 ;

2

;

₁

0

^;P1^;^"1^I

0 0 ;

2

0

^;P2

3

7

5

(21)

where

^P1

=

^P1

,

^P2

=

^P2

0, ;

1

=

^;

;

1

and ;

2

=

^;

;

2

. We do not require

^P1

to be positive semidenite, while

^P2

must be so in order not to violate

11

(

^j!

)

0 and

22

(

^j!

)

0. Since, (21) satises (20) for

=

1

^;^"2

1 +

^"2

] and

^j^j

_

^d

we can allow any time-varying convex-cone combination if the IQC holds. The variable

^"1

is an arbitrarily small positive constant that is chosen in order to allow a variation of

within small bands of width 2

^"2

around

1. To summarize, we have arrived at the following theorem:

Theorem 2 Assume that

ⁱ

and _^

ⁱ

satisfy (13) and that the interconnection of and

^G

(

^s

) is well-posed for every in the convex cone dened by

ⁱ

. Then the system

^u

=

^G

(

^s

)

^u

is asymptotically stable for any time-varying convex- cone combination of

ⁱ

and _^

ⁱ

if there exists a as dened by (12) such that

2

4

G

(

^j!

) 0

I

0

^I

3

5

(

^j!

)

2

4

G

(

^j!

) 0

I

0

^I

3

5

<

0 (22a)

11

(

^j!

)

0 (22b)

22

(

^j!

)

0 (22c)

hold for all

^!²^R^f1g

.

(10)

3.3 Comparison with Previous Criterion

Jonsson and Rantzer proposed an IQC for the stability analysis of linear system with slowly varying parameters 3, 2]:

M

I

R R

+

^V^R^V^R

+

^d²^V^S^V^S

+

^U^S^U^S ^S^;^S

S ;S ;R

(

Î^;Û^RÛ^R

)

^R

M

I

<

0 (23) where

R

=

^D^R

+

^C^R

(

^sI^;^A^R

)

^;¹^B^R ^S

=

^D^S

+

^C^S

(

^sI^;^A^S

)

^;¹^B^S

U

R

=

^C^R

(

^sI^;^A^R

)

^;¹ ^U^S

=

^C^S

(

^sI^;^A^S

)

^;¹

V

R

= (

^sI^;^A^R

)

^;¹^B^R ^V^S

= (

^sI^;^A^S

)

^;¹^B^S

such that

^R

and

^S

are both stable. We will now show that (23) is more conservative than (10). Specically, we will show that it is slightly more conservative than theorem 2 even if we assume that ~

11

0 and ~

22

0. Let

~ =

2

6

4

I

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

^;I

0 0

0 0 0 0 0 0

^I

0 0 0

^I

0 0 0 0 0 0

0 0 0 0

^d²^I

0 0 0 0 0

^;I

0 0 0 0

0 0

^I

0 0 0 0 0 0 0

0

^;I

0 0 0 0 0 0 0 0

^;I

0 0 0 0 0 0 0 0 0 0

^;I

3

7

5

and

T

=

2

4 R

S

I 3

5

U

=

2

4 U

R

0 0

^U^S

0 0

3

5

V

=

V

R

V

S

:

This yields

2

6

4

TM

0

VM

0

T U

0

^I

3

7

5

~

2

6

4

TM

0

VM

0

T U

0

^I

3

7

5

=

2

6

4 3

7

5 2

6

4

R R

+

^V^R^V^R

+

^d²^V^S^V^S ^S^;^S

0

^U^S

S ;S ;R R ;R U

R

0

^;U^R^R ^;

(

^I

+

^U^R^U^R

) 0

U

S

0 0

^;I

3

7

5 2

6

4 M

0 0

I

0 0 0

^I

0 0 0

^I

3

7

5

<

0

^:

Using Schur complements, we rewrite it as

M

I

R R

+

^V^R^V^R

+

^d²^V^S^V^S

+

^U^S^U^S ^S^;^S

S ;S ;R

(

^I

+

^U^R^U^R

)

^;¹^R

M

I

<

0

^:

(24)

(11)

Note that (24) is very close to the IQC (23) proposed by Jonsson and Rantzer

3, 2] for slowly varying uncertainties, which is slightly more conservative, since

I;U

R U

R

(

^I

+

^U^R^U^R

)

^;¹^:

for all

^U^R

. Note that main restriction in (23) is that it assumes semidenite

~

22

and ~

11

.

4 Examples

We will show the stability criterion on two examples. In both examples the improvements compared to previous IQC-based bounds are signicant.

4.1 Example I

This example is from 3]. A simple SISO system is considered:

_

x

(

^t

) = (

^A

+

^C

(

^t

)

^B

)

^x

(

^t

) =

;

0

^:

21

^;

1 + 0

^:

8 (

^t

)

1 0

x

(

^t

)

where it is assumed that

(

^t

)

²

^;

]. In the case the uncertainty parameter is constant, the system is asymptotically stable if and only if

^<

1

^:

25, that is, when

^<

1

^:

25. If _

is unbounded, then stability is assured if

^jj ^<

0

^:

261. This bound is obtained as the inverse of the

^H¹

-norm of

G

(

^s

) =

^D

+

^C

(

^sI^;^A

)

^;¹^B

where

A B

C D

=

2

4

;

0

^:

21

^;

1 0

^:

8 1 0 0

0 1 0

3

5

:

We now apply theorem 2 using

T

(

^s

) =

"

1

s+1

1

#

:

(25)

This multiplier has the same pole as that in 3]. There the analysis using a frequency dependent IQC yields a stability bound of

^j^j

_

^<^d

= 0

^:

1 for

= 1.

Using (25) yields

^d

= 0

^:

1036, which is in close agreement with 3].

If we instead choose

T

(

^s

) =

"

1

s+10

1

#

(26)

we obtain

^d

= 0

^:

1456 for

= 1, assuming ~

11

0 and ~

0. Relaxing the

requirements on ~ and instead assuming

11

(

^j!

)

0 and

22

(

^j!

)

0, we

obtain

^d

= 0

^:

3111 for

= 1 with (26). Figure 2 shows the stability bounds,

^d

,

as a function of

. Instability has been veried for

= 0

^:

41 without any bounds

(12)

0 0.2 0.4 0.6 0.8 1 1.2 0

0.1 0.2 0.3 0.4 0.5

d

stable

Figure 2: Example I: Stability bounds

^d

as a function of

. The system is stable of

(

^t

)

²

^;

] and

^j

_ (

^t

)

^j ^d

. The lower, dashed line uses

^T

as dened in (25) while the middle, dashed-dotted line uses (26). Both assume semidenite

~

ⁱⁱ

. The upper, solid line uses the relaxed constraints on ~

ⁱⁱ

and the multiplier

dened by (26). The cross (

) marks the bound given in 3]. The circle (

)

at

= 1

^:

25, denotes the bound obtained with constant

. Arbitrarily fast

variations in

can be allowed if

^jj^<

0

^:

261, marked by the vertical line. The

criteria show stability in the area below and to the left of the lines. The asterisk

(

) at

= 1 and

^d

= 0

^:

46, shows a case for which instability has been veried.

(13)

on _

and

^d

= 0

^:

46 for

= 1 these bounds are (probably) not the true bounds though.

The lower bound could probably be improved by using a basis multiplier,

T

, of high order. However, this leads to an increased computational load, as well as that the condition number of the required ~ usually increases. This may cause numerical problems, which could result in worse bounds even if this is not theoretically true.

4.2 Example II

This example is has been analyzed in 4] using IQCs. We consider the ship steering dynamics as in Example 9.6 in 1]. The dynamics can be approximated by the Nomoto model

_

x

(

^t

) =

^v

(

^t

)(

^;ax

(

^t

) +

^bv

(

^t

)

(

^t

)) _

(

^t

) =

^x

(

^t

)

where

denotes the heading of the ship,

denotes the rudder angle and

^v

is the speed of the ship. It is assumed that

^v

(

^t

)

0. We will, as in 1], study the stability of the ship dynamics for an unstable tanker with

^a

=

^;

0

^:

3 and

^b

= 0

^:

8, which is controlled by a PD regulator

v

=

^;K

where

^K

(

^s

) =

^k

(1 +

^T^d^s

) with

^k

= 2

^:

5 and

^T^d

= 0

^:

86. The closed loop system can be described by

x

(

^t

) = (

^v

(

^t

)

^I2

)

^G0

(

^s

)

^x

(

^t

) with

G0

(

^s

) =

;a=s b

;K

(

^s

)

^=s²

0

:

This system is not strictly stable and instead we replace it for the purpose of analysis by

G

(

^s

) = (

^I

+

^G0

(

^s

))(

^I ^;^G0

(

^s

))

^;¹

=

^D

+

^C

(

^sI^;^A

)

^;¹^B

where

A B

C D

=

2

6

4

;

(

^a

+

^bkT^d

)

^;bk ^;

2

^;

2

^b

1 0 0 0

a

+

^bkT^d ^bk

1 2

^b

kT

d

k

0 1

3

7

5

and

(

^t

) =

^v

(

^t

)

^;^vnom

v

(

^t

) +

^vnom ²

^;

1 1]

where

^vnom

is the nominal speed of the ship. For the transformed system,

^d

is solved as a function of

such that the system is veried to be stable when

jj

and

^j^j

_

^d:

An IQC-Based Stability Criterion for Systems with Slowly Varying Parameters