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

A. 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

May 25, 1994

33rd CDC

Abstract

This paper discusses state-space methods for analyzing stability of continuous time linear systems subject to structured uncertain- ties. Four types of uncertainties are discussed: linear parametric and dynamic uncertainties (real and complex ) and nonlinear paramet- ric and dynamic uncertainties. The method employs LMIs equipped with a scaling matrix adapted to the type of uncertainty. For para- metric uncertainties conservativeness is reduced by branch and bound schemes. Dierent types of uncertainties can be mixed in this ap- proach. The problem is convex except for the linear dynamic (complex

) case.

Keywords:

Structured uncertainty, state space, quadratic stabil- ity.

1

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 6], the tools became more easy to use and the spread to new applications took a stride. Doyle 5] 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 a singular value. The minimization of the singular value can be reformulated to a linear matrix inequality (LMI). The

^

-analysis (complex and real) are normally performed in the frequency domain. First the transfer function is determined at a set of discrete frequencies suciently close to each other in order not to miss any signicant details (peaks). Then, the

^

-methods (see e.g. 14]) are employed at each of these frequencies. If each of these tests succeed the system will be stable subject to the uncertainties specied if the frequency coverage was dense enough.

This situation resembles very much the technique used for

^H1

norm computation prior to the state-space methods presented in 1988 2], 11].

As for the linear case of uncertainties, a nonlinear class of uncertainties can be analyzed in a framework similar to the linear case. Two types of nonlinear uncertainties are treated corresponding to the real and complex linear ones: nonlinear parametric and nonlinear dynamic uncertainties re- spectively. The extension from the linear

^

-analysis is quite natural and straight forward. The non-linear analysis is performed in the state-space domain and is based on a quadratic Lyapunov function and are sometimes denoted quadratic stability or

^Q

-stability analysis 10].

This paper discusses state-space methods for stability analysis of systems with mixed uncertainties of these kinds. Ways of reducing conservativeness with branch and bound schemes will also be indicated.

The report is outlined as follows: In section 2 criteria for assuring stability of continuous-time systems are given. These are formulated as linear matrix inequalities (LMIs). In section 3 systems with non-linear uncertainties are

2

treated by using constant scaling matrices. These are the most general ones and provides conservative bounds if less general uncertainties are analyzed with these methods. In section 4 dynamic scaling is applied for dealing with linear dynamic uncertainties (complex

^

). Branch and bounds methods for reducing conservativeness for parametric uncertainties, both linear (real



) and nonlinear are discussed in section 5. Means for handling arbitrary dependency on parameters are also suggested.

1.1 Notations

X

T

denotes the transpose of

^X



^{X >}

(

^

)0 a symmetric, positive denite (semidenite) matrix diag 

^X1X2

] a block-diagonal matrix composed of

^X1

and

^X2



^kxk2

the

^L2

-norm of the vector

^x

 

^

(

^X

) the maximal singular value of

^X

and

^k:k1

the

^H1

norm.

2 Linear Matrix Inequalities (LMIs)

2.1 Bounded Real Lemma

In this paper linear matrix inequalities (LMIs) are used for analyzing the stability of systems with uncertainties. We rst consider the problem of

nding the

^H1

-norm of a stable system

^G

dened by

^G

(

^s

) =

^D

+

^C

(

^{sI ;}

A

)

^;1B

. Then

^{kGk1 < }

is equivalent to the existence of a symmetric, positive denite matrix

^{P >}

0 satisfying

2

6

4

PA

+

^{ATP PB CT}

B T

P ;I D

T

C D ;I

3

7

5

<

0

^:

(1)

This inequality is called the Bounded Real Lemma, see e.g. 13], 7]. This inequality will be used in di erent shapes in this paper.

2.2 Solving LMIs

During the last years methods for solving LMIs have been developed and made available 3], 8]. Methods are still being elaborated for higher eciency and user exibility.

3

3 Structured Dynamic Uncertainties

The block diagram of system

^G

subject to structured uncertainties, linear or nonlinear, according to the

^

-formalism is depicted below:

G



- w

1

- z

1 -

w

2



z

2

The system subject to uncertainties is assumed to be linear and described

by

^x

_ =

^Ax

+

^B1w1

+

^B2w2

z

1

=

^C1x

+

^D11w1

+

^D12w2

z

2

=

^C2x

+

^D21w1

+

^D22w2:

(2) The structured uncertainties are described by the -block. In the

^

- formalism these are either complex or real, and bounded by the

^H1

norm.

For notational convenience we may assume that

^

= 1 by proper scaling of the system. In the nonlinear case we extend the class of uncertainties to include nonlinear dynamic elements bounded by

kw

2i k

2

kz

2i k

2

 i

= 1

^:::r:

(3) Also we assume for notational convenience that every -element is square, that is, it has equal number of inputs and outputs this can be achieved by introducing dummy inputs or outputs.

We now want to nd a criterion for showing that

^{kz1k2 <kw1k2}

subject to (3). This is satised if

kz

1 k

2

2

;kw

1 k

2

2

+

^Xr

i=1

i



k^z

^

^2ik22;kw

^

^2ik22<

0 (4) for some set of positive lagrangian multipliers

^{i >}

0.

4

Let  = diag

^{I 1I::: rI}

], where the blocks correspond to the struc- ture of the -elements. Then, (4) can be rewritten as

k



^{zk2 <k}



^wk2

(5) where  (denoted

^D

in

^

-formalism) is a scaling matrix corresponding to  satisfying 

^T

 = . In the following discussion we assume that  is a sym- metric, positive denite, block diagonal matrix,  = diag 

^0



^1:::



^r

]

^>

0, which is more general to what has been considered so far.

G

 

- -

?

?

?

?

w z

~

w ^z

~

By minimizing the

^H1

norm from ~

^w

to ~

^z

with respect to  (or ) we obtain a sucient criterion for stability. Inserting the scaled system in (1)

we obtain

²

6

4 A

T

P

+

^{PA PB CT}

B T

P ;



^DT

C D ;



^;1

3

7

5

<

0

^

(6)

which in turn can be rewritten into

"

A T

P

+

^{PA PB}

B T

P ;



#

+

"

C T

D T

#



^{h C D i<}

0

^:

(7) Note that this inequality is linear in (

^P

) and can be solved by LMI algo- rithms. Also, the set of solutions (

^P

) is convex. It might also be interrest- ing to note that there is an input-output duality. An equivalent formulation can be obtained by replacing (

^ABCD

) and (

^P

) by (

^ATCTBTDT

) and (

^P;1



^;1

) respectively.

The -matrix corresponds to the scaling matrix used in the

^

-analysis for computing an upper bound of the

^

-norm of a system. In the linear

5

dynamic case the scaling matrix is frequency dependent and the

^

-norm is computed by sweeping over the relevant frequency range and computing a scaling matrix and the

^

-norm for each frequency. In the nonlinear case the scaling matrix, , becomes constant (not frequency dependent). Loosely speaking, we have that  (or ) and  shall commute ( = ). For the nonlinear case a constant  is required since energy can be transformed from one frequency to another. The fact that  is constant implies that the Riccati inequality can be solved in the state-space domain. The nonlinear analysis gives a more conservative result than the linear

^

-analysis.

4 Complex -analysis

Usually the

^

-analysis is performed in the frequency domain by sweeping over the frequencies of interest and computing the

^

-norm. For complex uncertainties the upper bound is determined by



(

^M

) = inf

D





(

^DMD;1

)

As for the

^H1

norm computed by a frequency sweep there is a risk that the worst frequency is missed. An alternative to the frequency-sweep method is to perform the analysis in the state-space domain. In this case, the

^D

-matrix is replaced by a dynamic system, given by ^

^G

(

^s

) = ^

^D

+ ^

^C

(

^{sI ;A}

^ )

^;1B

^ . By a proper choice of ^

^G

, we can prove that the

^

-norm for the system

^G

(

^s

) =

D

+

^C

(

^{sI ;A}

)

^;1B

is less than 1, by showing that:

k^zk

^

^{2 <kwk}

^

²

where ^

^w

= ^

^G

(

^s

)

^w

and ^

^z

= ^

^G

(

^s

)

^G

(

^s

)

^w

for all possible inputs

^w

. The struc- ture is depicted below:

G

^

G ^G

^

- -

?

?

?

?

w z

^

w ^z

^

6

Writing this in state-space form we get

d

dt 2

6

4

^

x x

^

1 x

2 3

7

5

= ~

^A

2

6

4

^

x x

^

1 x

2 3

7

5

+ ~

^Bw

~

A

=

2

6

4

A

0 0

0 ^

^A

0

^

BC

0 ^

^A

3

7

5

~

B

=

2

6

4 B

^

^

B BD

3

7

5

^

w

= ^

^Cx

^

¹

+ ^

^Dw

^

z

= ^

^Cx

^

²

+ ^

^Dz

= ^

^Cx

^

²

+ ^

^DCx

+ ^

^DDw:

The related LMI can be written as

"

~

A T

P

+

^PA

~

^PB

~

~

B T

P

0

#

+

2

6

6

6

6

4

( ^

^DC

)

^T

^ 0

C T

( ^

^DD

)

^T

3

7

7

7

7

5

h ^DC

^ 0 ^

^{C DD}

^

ⁱ

; 2

6

6

6

6

4

^ 0

C T

^ 0

D T

3

7

7

7

7

5

h

0 ^

^C

0 ^

^{D i<}

0

^:

By introducing

 =

"

^

C T

^

D T

#

h ^C

^

^D

^

^i>

0 we can rewrite the LMI into

"

~

A T

P

+

^PA

~

^PB

~

~

B T

P

0

#

+

2

6

6

6

4

0

^CT

0 0

I

0 0

^DT

3

7

7

7

5



"

0 0

^I

0

C

0 0

^D

#

7

; 2

6

6

6

4

0 0

I

0 0 0 0

^I

3

7

7

7

5



"

0

^I

0 0 0 0 0

^I

#

<

0

^:

(8) As we can see the LMI is on the standard form if ^

^A

and ^

^B

are given, that is, it is linear in

^P

and . Thus the optimal ^

^C

and ^

^D

can be determined by the LMI-algorithm. Then, if

^P

is kept constant, we may improve on ^

^A

, ^

^B

and  since the inequality is linear in these. Note that the structure of ^

^AB

^

^C

^

^D

^ , and consequently , depends on the structure of the uncertainties.

We can now propose the following algorithm for nding a scaling system

^

G

satisfying a given value of

^

. We assume that

^P

is structured as

P

=

2

6

4 P

11 P

12 P

13

P T

12 P

22 P

23

P T

13 P

T

23 P

33 3

7

5

:

(9)

(i) Start with some ^

^AB

^ 

(ii) Keeping ^

^AB

^ constant, solve the LMI (8) with respect to

^P

and 

(iii) Keeping

^{P12P13P22P23P33}

constant, solve the LMI (8) with respect to

^P11A

^

^B

^ and 

(iv) Iterate (ii) and (iii) until the required value of

^

is achieved or until

^

cannot be reduced any longer.

Note that

^B

,

^C

and

^D

are functions of

^

. In step (i) we could use an initial solution obtained from the frequency domain

^

-analysis. Using the

^

- Analysis and Synthesis Toolbox 1] with Matlab this can be performed using the

^mu

and

^musynfit

commands.

8

5 Structured Parametric Uncertainties

5.1 Introduction

A sucient condition for assuring stability for a set of systems

^S

, is that there exists a pair (

^P

), such that

"

A T

P

+

^{PA PB}

B T

P ;



#

+

"

C T

D T

#



^{h C D i<}

0 (10) is satised for every (

^ABCD

)

^2S

. If this condition is satised,

^S

is said to be quadratically stable (

^Q

-stable), since it involves a quadratic Lyapunov function

^V

(

^x

) =

^xTPx

.

Parametric nonlinear uncertainties can be treated in a similar way as the dynamic uncertainties. We dene a parametric uncertainty as

w

i

=

ⁱ

(

^t

)

^{zi i}

(

^t

)

²



^;

1

^

1]

^:

(11) Comparing this with the dynamic uncertainty

^{kwik  kzik}

, it is clear that stability for dynamic uncertainty implies parametric stability.

The parametric uncertainty

ⁱ

may be time dependent. This includes any dependency on the states or any input, both exogenous and endogenous. In order to include this in the machinery developed in section 4, we extend the class of scaling matrices, 

ⁱ

, to any symmetric, positive denite matrix, that is, 

^{i >}

0. The Riccati inequality will provide a sucient condition, since

w T

i



ⁱ

(

^t

)

^wi

=

ⁱ

(

^t

)

^2ziT



ⁱ

(

^t

)

^{zi ziT}



ⁱ

(

^t

)

^zi:

(12) If we restrict 

ⁱ

to be constant it can be included in the same formalism as was developed in section 4. This will, however, provide a conservative bound for a parametric uncertainty.

In 9] it is shown that in the case of one block of uncertainty, parametric and dynamic uncertainty are equivalent. In the case of two or more blocks this equivalence does not hold, see for example 12].

We will now look at two methods of analysis and design when parametric uncertainties are involved. In the rst method we assume that

^Dii

= 0, which allows us to check the vertices only of the parameter space. In the general case, which also can be used for design, we apply a branch and bound scheme.

By dividing the parameter space into subregions, in which the uncertainties are treated as dynamic, we can obtain performance bounds that are less conservative.

9

5.2 Checking the Vertices of the Parameter Space

In order to assure that it is sucient to only the vertices of the parameter space (in

) we must assume that

^Dii

= 0. Here, we let

^Dii

denote the

^D

matrix corresponding to (all of the) nonlinear parametric uncertainties. If

D

ii

=

^diiI

, this can be transformed to an equivalent state-space representa- tion with

^Dii

.

The requirement on

^Dii

probably requires some comments. First, in order to only check the vertices of the the parameter space it is necessary that all

D

-elements related to these parameters are zero, that is not only the

^Dii

elements in each block but also the

^Dij

elements connecting the di erent parameter blocks. However, the other

^D

elements may be non-zero.

Secondly, this requirement is not too conservative since it is usually the case when we start from the other end by dening the set of systems as a convex set in (

^ABCD

) dened by a number of extreme systems, which can be considered as vertices in the parameter space. Thus,

^Dii

becomes zero.

5.3 Branch and Bound Schemes

In the general case we may employ the time-dependent freedom in  and compute a constant  for subregions of the parameter space. By dividing the parameter space in subregions the conservativeness will be reduced when treating parametric uncertainties as dynamic. Especially, in the limit we get

(

^t

) = (

(

^t

)) as a (continuous) function of

.

Branch and bound schemes has been used in real

^

-analysis for improving the accuracy of the stability bound, see e.g. 4], 14].

Another advantage with the branch-and-bound scheme is that it o ers a way to treat quadratic stability in a less conservative way, when parame- ters enters non-linearly in the system. Suppose that a system depends on a parameter



and that elements of (

^ABCD

) in the state-space represen- tation depend on



and

²

linearly. Assume for notational convenience that

 2



^;

1

^

1]. Then introduce uncertainties

¹

=



and

²

= 2

^2;

1. We start by the conservative assumption that these uncertainties are independent and determine the performance of the system that way. In order to improve the performance bound, we split the parameter space in two parts: A) and B), see Figure 1.

10

-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

rho_1 = theta

rho_2 = 2*theta^2 - 1

parameter space

A B

Figure 1: Splitting the parameter space into two subregions: A and B By dividing the parameter space further, the delity can be increased and the conservativeness can be reduced arbitrarily. However, since we still are conned to quadratic stability (

^V

(

^x

) =

^xTPx

) a certain amount of conser- vativeness will always remain in the general case.

5.4 Nonlinear Parametric Uncertainties

When nonlinear (time-varying) parametric uncertainties are to be analyzed, the parameter space is divided into subregions, usually by bisecting one or several of the parameter intervals. Thus, a set of LMIs as dened by (7) emerges. These LMIs are coupled by a common

^P

matrix and possibly com- mon block in the  matrix corresponding to dynamic uncertainties. Blocks in the  matrix corresponding to parametric uncertainties can take di erent values in each parameter subregion. This exploits the fact that the  block can be time-dependent and, specically, it can depend on the parametric uncertainty itself.

11

5.5 Linear Parametric Uncertainties (Real )

Linear parametric uncertainties (real

^

analysis) is treated in a similar way as the nonlinear case. The parameter subspace is divided into subregions for each of which an LMI is solved. For linear uncertainties the set of LMIs will be completely decoupled, that is both the

^P

and  matrices may be di erent.

5.6 Pruning

In order to reduce the computational burden and arrive at a tractable scheme it is necessary to reduce the number of subregions by some pruning method.

This can be done, for instance, by nding upper and lower bounds of the gain

^

for each subregion. The upper bound is obtained by considering the parametric uncertainty as a dynamic nonlinear one. The lower bound can be computed by considering the parameters as constant and computing the gain in the vertices of the regions. If the upper limit in one subregion is less than the lower limit of any other subregion the rst subregion do not need to be analyzed any more. By increasing the number of subregions the di erence between the upper and lower limits is reduced and the accuracy of the computation is increased accordingly.

6 Conclusions

Methods for analyzing mixed linear and nonlinear uncertainties both para- metric and dynamic have been discussed. Dynamic uncertainties are treated by using constant scaling matrices in the nonlinear case and dynamic scaling systems in the linear (complex

^

) case. The nonlinear problem is convex. In the complex

^

case the problem is not generally convex and (global) conver- gence is not guaranteed.

Parametric uncertainties are based on the same framework as is used for dynamic nonlinear uncertainties. In order to reduce conservativeness the scaling block are allowed to be any symmetric positive denite block.

Also, further improvement is obtained by splitting the parameter space into subregions (branch and bound). Both parametric problems are convex since they are based on the nonlinear dynamic case.

Note that all four cases of uncertainties can be combined in the analysis.

12

The picture below shows the relation between the di erent methods re- lated to the uncertainty classes. The conservativeness is reduced as we move downwards.

Nonlin. dynamic

Real

^

Complex

^

Nonlin. parametric

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^

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15