Applications of mixed -synthesis using the passivity approach

(1)

Applications of mixed -synthesis using the passivity approach

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

September 26, 1994 1995 ECC

Abstract

This paper presents mixed synthesis using a

^D

-

^K

like method in- spired by passivity theorem. The method is illustated on two similar examples showing the virtues of the mixed approach.

Keywords:

Structured uncertainty, mixed mu synthesis, passivity.

1 Introduction

Structured singular values ( ) is used for analysis and synthesis of systems with uncertainties or perturbations 1, 6]. Complex uncertainties representing dynamic uncertainties can be treated in the synthesis procedure using the D - K iteration scheme. Even if this algorithm is not guaranteed to converge to the global minimum, it provides a good design in most applications. Recently also real (parametric) uncertainties have been included in the synthesis procedure

4, 7].

In this paper a design method based the positive real Parrot theorem 4] is used. Using the scaling matrices from the analysis we can obtain multipliers that make the system positive real. Since the positive real property can be translated to an H

¹

problem, this ts neatly to the H

¹

synthesis scheme using sector transformations.

The paper is outlined as follows: In section 2 the analysis is reviewed

mainly focusing on the computation of the upper bound. This section is fairly

standard. The algorithm for mixed synthesis is given in section 3, and it is

illustrated by two examples in section 4. Conclusions are given in section 5.

(2)

1.1 Notations

We de ne X

( s ) = X ^T (

^;

s ), X ^T denotes the transpose of X X > (

)0 a sym- metric, positive de nite (semide nite) matrix diag X

¹

X

²

] a block-diagonal matrix composed of X

¹

and X

²

^k

x

^k²

the L

²

-norm of the vector x ( X ) the maximal singular value of X

^k

:

^k¹

the H

¹

norm sect( X ) = ( I

^;

X )( I + X )

^;1

denotes the sector transformation and herm( X ) =

¹²

( X + X

).

2 -analysis

This section gives a short review on structured singular values and sector trans- formations (LFTs), see also e.g. 2].

2.1 Denitions

We will here adopt the same notation and de nition as is used in 8, 9]. The de nition of the function depends upon the underlying block structure

^K

of uncertainties , which could be either real or complex, see below. 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.

M

-

Given a matrix M

²^C

ⁿ

ⁿ and three non-negative integers m r , m c and m C

with m = m r + m c + m C

n the block structure is an m -tuple of positive integers

K

= ( k

¹

::: k m r k m r

⁺¹

::: k m r

⁺

m c k m r

⁺

m c

⁺¹

::: k m r

⁺

m c

⁺

m C ) (1) where

^P

^m _i

⁼¹

k i = n for dimensional compatibility. The set of allowable per- turbations is de ned by a set of block diagonal matrices

^X^K ² ^C

ⁿ

ⁿ de ned by

X

K

=

^f

= diag

¹

^r I k

¹

::: _rk _mr I k _mr

^c

¹

I k _mr

⁺¹

::: _cm _c I k _mr

⁺

_mc ^C

¹

::: _Cm _C ] :

_ri

²^R

_ci

²^C

_Ci

²^C

^k ^mr

⁺

^mc

⁺

ⁱ

^k ^mr

⁺

^mc

⁺

ⁱ

^g

: (2) The structured singular value of a matrix M

²^C

ⁿ

ⁿ is de ned by

K

( M ) =

min

2X

K

f

() : det( I

^;

M ) = 0

^g

;1

(3)

and if no

²^X^K

satis es det( I

^;

M ) = 0 then

^K

( M ) = 0.

(3)

2.2 The Upper Bound

Generally the structured singular value cannot be exactly computed instead we have to resort to upper and lower bounds, which are usually sucient for most practical applications. A tutorial review of the complex structured singular value is given in 5].

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

. For complex uncertainties the upper bound is determined by

^K

( M ) = inf _D

2DK

( DMD

^;1

) (4)

where

^D^K

is the set of block-diagonal, positive de nite Hermitian matrices that commute with

^X^K

, that is

D

K

=

^f

0 < D = D

²^C

ⁿ

ⁿ : D = D

⁸

²^X^K^g

: (5) This problem is equivalent to an LMI problem

^K

( M ) = inf _>

P

^2DK0

f

: M

PM <

²

P

^g

: (6) The mixed uncertainties (real and complex) can be included in the LMI problem for computing the upper bound (see e.g. 3, 8, 9]). We rst de ne

G

K

=

^f

G = G

²^C

ⁿ

ⁿ : G =

G

⁸

²^X^K^g

(7)

Theorem 2.1 Let

^K

( M ) = inf _>

P G

^2D^2GK0^K

: M

PM + j ( GM

^;

M

G ) <

²

P

(8) with

^D^K

and

^G^K

as dened in (5) and (7) respectively. Then

^K

( M )

^K

( M ).

Note that G

²^G^K

is block diagonal with zero blocks for complex uncertain- ties. If we let G = 0 in (8) we recover the complex upper bound (6).

We can also reformulate the inequality into a form that corresponds to a real posivite test. Let

W

K

=

^f

W = P + jG : P

²^D^K

G

²^G^K^g

(9) The multiplier W is positive real (herm( W ) > 0, herm W =

¹²

( W + W

)) it has symmetric blocks for complex uncertainties and full (not necessarily symmetric) blocks for real uncertainties. We then have

^K

( M ) = inf _>

W

^2WK0

f

: herm

^;

( I +

¹

M )

W ( I

^;¹

M )

> 0

^g

(10) By pre- and post-multiplying by ( I +

¹

M )

^;1

and ( I +

¹

M )

^;

, respectively, we obtain

^K

( M ) = inf _>

W

^2WK0

f

: herm( W sect(

¹

M )) > 0

^g

(11) Comparing this with (8), we have P =

¹²

( W + W

) and G = j

¹²

( W

^;

W ), and consequently W = P +

¹

jG . Now assume that W is factorized: W = Y

Z . Then (11) can be rewritten as

f

: herm( Z sect(

¹

M ) Y

^;1

) > 0

^g

(12)

(4)

2.3 Reformulation of the upper bound

The upper bound can be reformulated in a number of ways, which is important when nding ecient algorithm for computing it. The following theorem is from

9]. Theorem 2.2 Assume that > 0 and denote M D = DMD

^;1

. Then, the following statements are equivalent.

(i) There exist matrices P

¹²^D^K

and G

¹²^G^K

, such that

M

P

¹

M + j ( G

¹

M

^;

M

G

¹

) <

²

P

¹

(13) (ii) There exist matrices D

²²^D^K

and G

²²^G^K

, such that

M _D

²

M _D

²

+ j ( G

²

M _D

²^;

M _D

²

G

²

) <

²

I (14) (iii) There exist matrices D

³²^D^K

and G

³²^G^K

, such that

^;

¹

M D

³^;

jG

³

( I + G

²³

)

^;¹²

< 1 (15) (iv) There exist matrices D

⁴²^D^K

and G

⁴²^G^K

, such that

( I + G

²⁴

)

^;¹⁴^;¹

M D

⁴^;

jG

⁴

( I + G

²⁴

)

^;¹⁴

< 1 (16)

Proof: The equivalence between (i) and (ii) is obtained by replacing D

²

D

²

= P

¹

and G

²

= ( D

²

)

^;1

G

¹

D

^;1²

. Then we rewrite (ii) into

M D

²

^;

jG

²

M D

²

^;

jG

²

< I + G

²²

²

⁼

I + G

²²

²

1

2

I + G

²²

²

1

2

: (17) Thus,

I + G

²²

²

; 1

2

M D

²

^;

jG

²

M D

²

^;

jG

²

I + G

²²

²

; 1

2

< I (18) from which the equivalence between (ii) and (iii) follows with D

³

= D

²

and G

³

= G

²

= . Finally, by letting D

⁴

= ( I + G

²³

)

¹⁴

D

³

the equivalence between (iii) and (iv) follows by noting that G

³

( I + G

²³

)

¹⁴

= ( I + G

²³

)

¹⁴

G

³

.

²

To summarize we have the following relations between the scaling matrices in theorem 2.2.

statements D G

(i) - (ii) D

²

D

²

= P

¹

G

²

= ( D

²

)

^;1

G

¹

D

^;1²

(ii) - (iii) D

³

= D

²

G

³

= G

²

=

(iii) - (iv) D

⁴

= ( I + G

²³

)

¹⁴

D

³

G

⁴

= G

³

We also have

W = D

⁴

( I + G

²⁴

)

^;¹⁴

( I + jG

⁴

)( I + G

²⁴

)

^;¹⁴

D

⁴

(19)

Remark 2.1 For ! = 0 and ! =

¹

we observe that G ( j! ) = 0, since M ( j! )

is real.

(5)

3 Algorithm

3.1 The

^W

-

^K

iteration

The algortihm used is similar to the D - K iterations used for complex uncertain- ties and the D , G - K iteration scheme 7]. Instead of using symmetric scalings D , we use asymmetric multipliers, Y and Z , and the passivity theorem. This approach is a natural extension of the D - K iteration.

In each iteration the following step are taken.

(i) Find a W by tting a transfer function of the corresponding block struc- ture to D and G data from the analysis step. In the rst iteration, when no such data are available, choose W = I .

(ii) Factorize W = Y

Z into stable and minimum phase factors Y and Z . (iii) Determine an H

¹

controller K for the system sect( Z sect(

¹

M ) Y

^;1

) for

the smallest possible , for instance by bisectional search.

(iv) Perform a mixed- analysis on the closed loop system

^F

l ( MK ) and com- pute D , G and as functions of frequency.

(v) Repeat iterations until obtains target value or converges

In (ii) we have augmented Y and Z with a diagonal block corresponding to con- trol inputs and outputs. For notational convenience, we have assumed (without restriction) that the number of inputs are equal to the number of outputs.

3.2 How to obtain the multiplers

^Y

and

^Z

We will now look at the problem of nding a pair ( YZ ) that corresponds to ( DG ). According to (19) we have

Y

Z = W = D

⁴

( I + G

²⁴

)

^;¹⁴

( I + jG

⁴

)( I + G

²⁴

)

^;¹⁴

D

⁴

(20) where we require W to be positive real.

We assume that W can be represented as a proper rational transfer function matrix with no poles or zeros on the imaginary axis. We now want to factorize W = Y

Z , such that Y and Z are both proper, stable and minimum phase (inversely stable). In the general rational matrix case it is also possible to factorize any W in such a way.

In our case we let

Y = A

^;

D

⁴

Z = A

⁺

D

⁴

where

A

⁺

A

^;

= A = ( I + G

²⁴

)

^;¹⁴

( I + jG

⁴

)( I + G

²⁴

)

^;¹⁴

(21) where A

⁺

= A

^;1^;

stable and minimum phase. Both Y and Z have two factors:

one amplitude part D

⁴

and one phase part A

^;

or A

⁺

. Note that A is all-pass.

(6)

When using the ^mu -command in the -Analysis and Synthesis toolbox, we obtain two scaling matrices: D ( ! ), which is a nonsingular matrix, and G ( ! ), which is diagonal and real. For complex uncertainties we have G = 0. These scaling matrices are functions of ! in a discrete number of points, usually loga- rithmically distributed.

We restrict the discussion to the scalar block case. First, we nd a D

⁴

that ts D ( ! ) such that

^j

D

⁴

( j! )

^j^j

D ( ! )

^j

. Here we let

denote an approximation using some tting objective, possibly associated with a weighting function with respect to frequency. This can be performed using the ^musynfit or ^musynflp command.

Next, we determine an all-pass, positive real, approximation A that ts the phase of I + jG ( ! ), such that arg A ( j! )

arg( I + jG ( ! )) = atan G ( ! ).

3.3 Fitting

^G⁴

We start by observing that it is sucient to study the scalar case since any G is diagonal. Hence, each (diagonal) element in G can be treated separately.

It can be shown that G (0) = 0 and G (

¹

) = 0, see Remark 2.1. Then the all-pass function A must have a structure of the form

A ( s ) = 1

^;

a

¹

s + :::

^;

a

²

_N

^;1

s

²

^N

^;1

+ a

²

_N s

²

^N

1 + a

¹

s + ::: + a

²

N

^;1

s

²

^N

^;1

+ a

²

N s

²

^N ⁽²²⁾ The aim is now to nd an A such that

arg A ( j! )

atan G ( ! ) (23) and herm A > 0 or max !

^j

arg A ( j! )

^j

<

²

.

Equivalently we may approximate A by

A ( j! )

(sect(

^;

jG ( ! )))

¹²

(24) This can be formulated as a least square problem: minimize

X

i w _i

^j

(sect(

^;

jG ( ! _i )))

¹² ^;

A ( j! _i )

^j²

(25) with respect to a i and herm A > 0. Here w i is a sequence of weights.

Note that the all-pass approximation contains both stable and unstable poles and zeros. The stable part is moved to A

^;

and the unstable part is kept in A

⁺

. An example of a phase approximation using an all-pass lter is given in Figure 1. The phase must be constrained between

90 deg.

4 Examples

We will here illustrate the algorithm on two examples applied to the problem

of controlling and stabilizing the attitude of rocket using thrust vector control.

(7)

10

⁻¹

10

⁰

10

¹

10

²

−100

−80

−60

−40

−20 0 20

frequency [rad/s]

phase [deg]

Example of an all−pass approximation

atan G __

approximation − −

Figure 1: Example of a phase approximation using an all-pass lter. In the gure the solid line represents atan G , while the dashed line gives phase of the all-pass lter.

4.1 Example 1

The following requirements are used in this design example:

(i) a

²

0 1]

(ii) The gain margin shall be 6 dB or better

(iii) The phase margin shall be 35 degrees or better

(iv) The compensator gain shall be less than -6 dB at frequencies above 50 rad/s.

A state-space model is de ned by dt d

x

¹

x

²

=

0 1 a

⁰

0 x

¹

x

²

+

0 0 a

¹

0 w

¹

u

(26)

z

¹

y

=

1 0 1 0

x

¹

x

²

(27) where a

⁰

= 0 : 5 and a

¹

= 0 : 5.

A delay of 0.06 seconds is included for modeling computational delay, sam- pling eects and actuator dynamics. This delay is implemented by a rst order Pade approximation.

d ( s ) = 1

^;

0 : 03 s

1 + 0 : 03 s ⁽²⁸⁾

which is valid with a relatively good accuracy up to about 30 rad/s.

(8)

iter D + A multipliers H

¹

C R remark

1 - 2.5715 2.4620 - complex

2 3, 3, -] 1.1000 1.1000 1.1000 complex 3 3, 3, -] 1.0487 1.0487 1.0487 complex 4' 3+2 3, -] 0.9980 1.1222 1.0024 mixed 5' 3+2,3, -] 0.9976 1.1227 1.0015 mixed

Table 1: Example 1: summary of iterations

The gain and phase margins are assured by including a complex uncertainty in the feedback loop by the gain

1 + k m

²

p

1

^;

k _m

²

(29)

with k m = 0 : 6 and

^j

²^j

< 1.

For ensuring low enough gain at high frequencies the compensator gain is restricted by

^j

K ( j! )

^j

<

^j

W _k ( j! )

^j

where

W k ( s ) = 2 ( s + 7 : 436)

²

+ 19 : 746

²

( s + 7 : 072)

²

+ 49 : 841

²

( s + 20 : 537)

²

+ 9 : 357

²

( s + 60 : 285)

²

+ 72 : 894

²

(30) This requirement comes from the fact that the rocket is not a rigid structure but has exible modes due to the elasticity in structure and interstage joints.

In a conservative design, like this one, we make no assumption on phase since 50 rad/s is higher than the bandwidth of the system (determined by the total delay in the loop). Thus, stability is only assured by the gain requirement for

! > 50 rad/s.

The system is augmented by the a -variation (

¹

), the gain and phase margin requirement (

²

), and the high-frequency gain requirement (

³

).

4.1.1 Complex design

We start by treating all uncertainties as complex. The summary of the D - K synthesis is given Table 1. The column denoted D + A multipliers, gives the order of the scaling matrix D for

¹

,

²

and

³

respectively. The scaling for

³

is always one.

The D - K iteration converges after three to four iterations. We select the compensator from the third iteration since this is of slighty lower order than the fourth iteration compensator without any signi cant loss in performance. The obtained compensator with 19 states was reduced to a fth order compensator with a complex value of 1.0559.

4.1.2 Mixed design

By introducing asymmetric scaling ( Y and Z ) we can improve the real value to below 1, see Table 1. We start by using the compensator from iteration 3.

The scaling matrices were obtained by premultiplying D by the phase factors.

First and second order phase factors where used.

(9)

iter D + A multipliers H

¹

C R remark

1 - 9.4897 9.0640 2.0174 complex

2 1, 1, 1, -] 3.4995 2.1088 1.5942 complex 3 1, 1, 1, -] 1.9141 1.9140 1.9140 complex 3' 1+2, 1+3, 1, -] 1.1361 3.2773 1.1361 mixed 4' 1+1, 1+1, 1, -] 0.9700 4.2067 0.9729 mixed

Table 2: Example 2: summary of complex and mixed iteration. The order of the scaling matrices are given in the in the bracket list under the D + A multiplier column. For instance 1+2 denotes that D has rst order and A

^;

second.

Performing the same model reduction on the 5' iteration compensator we obtain a fth order compensator with = 1 : 016.

In this example very little is gained by treating the parametric uncertainty corresponding to a as real instead of complex. The two compensators are similar but the one obtained from the real design is slightly better.

4.2 Example 2

We will now try to extend the example by adding an explicit structural mode to the dynamics. In the previous example we had the requirement that the gain at frequencies above 50 rad/s should be less than -6 dB or 0.5. This requirement was set in order to not excite structural modes, of which the rst one was supposed to have a frequency of 50 rad/s or higher. We still make this assumption but we will add a new structural mode close to 10 rad/s. This means that the vehicle is structuraly weaker than before. The example given here does not give a fully realistic model of the vehicle dynamics, but serves the purpose of illustrating how the real synthesis method can be used for designing controllers where a complex design would fail.

We assume that the dynamics of the vehicle is described by the transfer function

G ( s ) = 1 s

²^;

a

^;

⁰ : 2

s

²

+ 2

¹

!

¹

s + !

¹²

⁽³¹⁾ where a

²

0 0 : 5] and

¹

= 0 : 01. Note that the requirement on a is relaxed since the vehicle has reduced bandwidth due to the structural mode at ! = 10 rad/s.

The frequency of the mode !

¹

is nominally 10 rad/s but we should try to design a compensator that allows as large variations as possible around this nominal value. We would like to achieve at least 10%, if possible, without sacrifycing the other requirements.

The augmented system has four uncertainties:

¹

for uncertainty in a ,

²

for !

¹

,

³

for gain and phase margin, and

⁴

for assuring low enough gain at

! > 50.

During the design procedure, see Table 2, we rst try to use complex

design. By doing this we can squeeze the value down to below 2, but not very

much further. One interesting observation is that if the real value is used as

criterion the compensator from the second iteration is better than the one from

(10)

10⁻¹ 10⁰ 10¹ 10² 10⁻¹

10⁰ 10¹ 10²

frequency [rad/s]

gain

Compensator using mixed mu design

Figure 2: Example 2: gain of compensator obtained using mixed design.

The compensator for the extended example with an explicit structural mode is given by a solid line. As a comparison the compensator from example 1 is also included (dashed line).

the next iteration. This is due to the fact that the complex synthesis does not exploit the full structure of the mixed problem.

In order to improve the design we start by using the second-iteration com- pensator for nding scaling matrices Y and Z . These are derived from the D and G scalings obtained from the real analysis. First a D is determined using the ^musynfit command. Then an all-pass lter A is determined by tting its phase to tan( G ). We then split the all-pass lter into two factors: one stable and minimum phase part A

^;

and a second antistable and anti-minimum phase part ( A

⁺

= A

^;1^;

, such that A = A

^;

A

⁺

).

The compensator obtained in the fth iteration has 39 states. The com- pensator is reduced to 9th order using Hankel norm reduction combined with explicit reduction of resonant pole-zero pairs. The reduced compensator yields a real- value of 0.9918, which ful lls the requirements.

The gain of the compensator is given in Figure 2. There is a marked notch in the gain at 10 rad/s. The main reason for this notch is to adjust the phase of the compensator in order to improve the robustness for variations in !

¹

. This twist in phase is clearly seen in Figure 3. The Nyquist diagram given in Figure 4 illustrates that the requirements are ful lled.

4.3 Discussion

This second example shows the advantage of the mixed design. In the rst example the gain by introducing real uncertainties was only marginal. In the second example the improvement was signi cant and the complex compensator actually failed to even come close to the requirement, which was satis ed by the mixed compensator.

The improvement can be explained as follows. The system has a pair of reso-

nant, badly damped, but stable poles. If their frequency is treated as a complex

uncertainty, this means that in the worst case these poles can be unstable. In

(11)

10⁻¹ 10⁰ 10¹ 10²

−800

−700

−600

−500

−400

−300

−200

−100

frequency [rad/s]

phase [deg]

Compensator using mixed mu design

Figure 3: Example 2: phase of compensator obtained using mixed design.

−4 −3 −2 −1 0 1 2

−3

−2

−1 0 1 2 3

real

imaginary

Nyquist diagram for compensator using mixed mu design

a=0 a=0.25

a=0.5

Figure 4: Example 2: Nyquist diagram of the open loop using the mixed

compensator. The dashed-line disc corresponds to the 6 dB and 35 deg stability

margin. Note that positive feedback convention is used.

(12)

order to obtain a stable closed loop system, these poles must be actively con- trolled and stabilized. This requires that a portion of the bandwidth must be allocated for the stabilization of the resonant poles. This could become in con-

ict with other requirements, and, thus, leading to a design not satisfying the design objectives.

If we instead add the fact that the frequency of the resonance is unknown, but that the modes must still be stable, only the proper phase around the resonance must be controlled for assuring stability.

5 Conclusions

The passivity approach for treating mixed- uncertainties in a -synthesis prob- lem have been illustrated on two examples.

Even if the uncertainty is real (parametric) it can in many applications be treated as complex. This is illustrated in the rst example in this paper. The complex- design is less complex than the real or mixed one.

In some applications the real approach is essential for obtaining the re- quired performance. The second example illustrates this on a system with a resonant mode with uncertain frequency.

6 Acknowledgement

This work was supported by the Swedish National Board for Industrial and Technical Development (NUTEK), which is gratefully acknowledged.

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, 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.

3] 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.

4] J. Ly, M. Safonov, and F. Ahmad. Positive real Parrot theorem with appli- cation to LMI controller synthesis. In Proceedings of the American Control Conference, volume 1, pages 50{52, Baltimore, Maryland, June-July 1994.

5] A. Packard and J. Doyle. The complex structured singular value. Automat- ica, 29(1):71{109, January 1993.

6] M. Safonov. Stability margins of diagonally perturbed multivariables feed-

back systems. In IEEE Proceedings of the Conference on Decision and Con-

trol, 1981.

(13)

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

8] P. Young, M. Newlin, and J. Doyle. analysis with real parametric un- certainties. In IEEE Proceedings of the 30th Conference on Decision and Control, volume 2, pages 1251{1256, Brighton, England, December 1991.

Applications of mixed -synthesis using the passivity approach