robust synthesis

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-Synthesis with Matrix Valued Scalings-

Algorithms and Examples

Tomas McKelvey and Anders Helmersson

Department of Electrical Engineering

Linkping University, S-581 83 Linkping, SwedenWWW: http://www.control.isy.liu.se

Email: {tomas,andersh }@isy.liu.se

March 1999

R E G L ER T E K N I K

A U T O M A T IC C O N T R O L

LINKPING

Report no.: LiTH-ISY-R-2134Presented at the American Control Conference, Albuquerque, NM, 1997, pp 361-365

Technical reports from the Automatic Control group in Linkping are available by anonymous ftp at the addressftp.control.isy.liu.se . This report is contained in the pdf-le 2134.pdf .


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-Synthesis with Matrix Valued Scalings - Algorithms and Examples

Tomas McKelvey and Anders Helmersson

Dept. of Electrical Engineering, Linkoping UniversityS-581 83 Linkoping, Sweden,

Fax: +46 13 282622email: [email protected], [email protected].

AbstractApproximation of rational matrix functions plays an

important part in the -synthesis algorithm of robust con-trollers if the plant is subject to repeated uncertainties.Algorithms for approximation of rational matrices/factorsto data are reviewed and practical issues are discussed.The methodology is illustrated by the design of a robustgain-scheduling controller for a linear time-varying sys-tem.

1. IntroductionApproximation of rational matrix functions to given

data plays a fundamental role in most automatic controland signal processing applications. When designing ro-bust controllers using synthesis a key step is the task of nding a state-space realization of a scaling matrix givenat a discrete set of frequencies. In this contribution we dis-cuss algorithms for determining real-rational matrix func-tions which well approximates given data. We focus ontwo types of problems:

1) Approximation of a multivariable spectral factor

2) Approximation of a positive real matrix function

The rst problem appears in the D -K iterations [2, 3]when the system has complex repeated uncertainties andthe second problem arises when the uncertainties are realvalued leading to a robust design algorithm known as Y -Z -K iterations [8, 7].

The approximation problem is tackled in a multi-stepfashion. In a rst step an unconstrained approximationof a multivariable state-space model to data is found us-ing a frequency domain subspace identication method[9]. After factorization of the initial approximant the fac-tors are parametrized and a quadratic criterion based on

the squared sum of the residuals are minimized using aLevenberg-Marquard type optimization algorithm.

This work was supported in part by the Swedish Research Coun-cil for Engineering Sciences (TFR) and the Swedish National Boardfor Industrial and Technical Development (NUTEK), which is grate-fully acknowledged.

The multivariable approximation problem has previ-ously been discussed in [11] and in this paper we demon-strate the proposed technique by a robust control synthe-sis example.

The paper has the following structure. A short reviewof -analysis is performed in Section 2. The D -K algo-rithm for synthesis of robust controllers is described and ageneralized D -K algorithm for parametric uncertainties ispresented. In Section 4 the subspace based algorithm forapproximation of D scalings is presented. A small designexample which illustrates the presented techniques can be

found in Section 5 and the paper is concluded in Section 6.

1.1. NotationRL is the set of real-rational matrix functions with

no poles on the imaginary axis. RH = {X (s) : X (s)RL , X (s ) analytic in Re ( s) > 0} is the set of stablereal-rational matrix functions. X denotes the com-plex conjugate transpose of X ; X > ( ) 0 a hermi-tian ( X = X ) positive denite (semidenite) matrix;X = ( X ) 1 ; diag [X 1 , X 2 ] a block-diagonal matrixcomposed of X 1 and X 2 ; herm X = 12 (X + X ); A B de-notes the Redheffer star product; (X ) denotes the max-imal singular value of X and G ( s) = G( s)T .

2. -analysisThis section gives a brief review on structured singular

values, see also [4].

2.1. Uncertainty StructureThe denition of the function depends upon the un-

derlying block structure of the uncertainties [18, 19].The set of allowable uncertainties is dened by a set of

block diagonal matrices C n n . Each sub-block canbe repeated scalars (either real or complex) or full blocks.The structured singular value of a matrix M C n n is

dened by

(M ) = min

{() : det( I M ) = 0 } 1

and if no satises det( I M ) = 0 then (M ) = 0.


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2.2. The Upper BoundGenerally the structured singular value cannot be ex-

actly computed; instead we have to resort to upper andlower bounds, which are usually sufficient for most prac-tical applications. A tutorial review of the complex struc-tured singular value is given in [13].

If only complex uncertainties are involved, the upperbound, which we here denote , is determined by

(M ) = inf D D

(DM D 1 ) (1)

where D is the set of block-diagonal, positive denite Her-mitian matrices that commute with . This problem isequivalent to an LMI problem. Also real uncertainties canbe included, see e.g. [5, 18, 19].

3. The D -K IterationsThis section gives a brief overview of the D -K algo-

rithm [2, 3]. Consider the problem of nding a robustlystabilizing controller to an LTI system G subject to LTIuncertainties. The set of scalings D that commute with is in this case any LTI scaling D that have a spa-tial structure commuting with the structure of , thatis D = D . Specically this implies that = D 1 D ,

if D is restricted to stable and inversely stable LTI sys-tems. To nd a controller K we can use the followingtwo-step iterative algorithm [2, 3] until convergence.

--

D G D 1

K -

Figure 1: Structure of the D -K problem.

(i) Given a xed controller K , nd a stable and inverselystable LTI scaling D such that D (G K )D 1 isminimized.

(ii) Given a xed scaling D , nd a controller K such thatD (G K )D 1 is minimized.

In the rst iteration we nd a controller K using Hsynthesis on the unscaled system ( D (s) = I ). We re-

peat the above two steps until the performance, =D (G K )D 1 , converges or becomes less than a spec-ied value.

The D -K iterations can be generalized to also includereal uncertainties by using passivity arguments Y and Z scalings [7]. Other approaches can be found in e.g. [16, 17].

3.1. D -K Iterations with LFT GainScheduling

Gain scheduling controllers can be synthesized usinglinear fractional transformations (LFTs) and linear ma-trix inequalities (LMIs), see [12, 8]. This type of designcan be seen as a natural extension of H synthesis. Thecontroller is assumed to be parametrized with respect toa set of parameters that are available to the controller inreal time. These parameters are assumed to be bounded,but they may vary without bounds on their rate of change.

In order to reduce conservativeness for the case when

the gain scheduling parameters are constant or slowlyvarying, frequency dependent scalings and multipliers canbe used. In such a case, matrix valued scalings and multi-pliers must be used since the same parameter, , appearsas a repeated scalar uncertainty both in the system andthe controller.

3.2. Motivation for Matrix-Valued Scal-ings

In -synthesis using D -K or Y -Z -K iterations thereis a need for tting state-space transfer functions to fre-quency data. In the case when the uncertainties are non-repeated these scalings and multipliers are scalars and forinstance the musynfit command in the -Analysis andSynthesis Toolbox [1] can be used. When uncertainties arerepeated the scalings need to be matrix-valued in order tonot reduce conservativeness. In this paper we propose amethod for achieving this.

4. Rational Matrix ApproximationIn step (i) of the D -K algorithm a real-rational stable

and inversely stable matrix function must be tted to thefrequency data delivered by the -analysis step. This cor-responds to the problem of nding a spectral factor. In thespectral factorization problem data obeys W k = W k > 0and a spectral factor G (s ) is sought which minimizes

k

W k G( j k ) G ( j k ) 2 (2)

where G, G 1 RH , i.e. G is a stable and inverselystable real rational matrix. In order to allow a spectralfactorization a given a rational approximant W (s ) mustobey certain conditions.

Lemma 1 (Spectral factorization [6]) Assume that W (s) = W ( s ) > 0, W ( ) > 0 and W, W 1 RL .

Then there exist matrix functions G such that W (s) = G( s )G (s) and G, G 1 RH .

In the generalized algorithm, Y -Z -K iterations a pos-itive real matrix function is tted to positive real data i.e.W k + W k > 0 and two factors Y (s) and Z (s ) are sought

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such that

k

W k Y ( j k ) Z ( j k ) 2 (3)

is minimized and Y , Y 1 , Z, Z 1 RH . The existenceof such a factorization is based on positive realness.

Lemma 2 (Positive real factorization [8, 6])Assume that W ( j ) + W ( j ) > 0, R {} and W, W 1 RL . Then there exist matrix functions Y, Z ,such that W (s ) = Y ( s )Z (s ) and Y, Y 1 , Z , Z 1 RH .

Constructive state-space algorithms for the factorizationsin lemma 1 and 2 can be found in [6] or [20].

4.1. Subspace Based Approximation Al-gorithm

In this section we discuss approximation of rationalmatrices from sampled data using a subspace based algo-rithm which comprises the rst step in the basic approxi-mation algorithm.

Consider a proper real-rational matrix function W (z)with p rows and m columns 1 . Any such matrix can bedescribed using a state-space realization

W (z) = C (zI A) 1 B + D. (4)

where A R n n , B R n m , C R p n and D R p m .If n is smallest possible the realization is known as mini-mal.

An algorithm which determines a state-space realiza-tion of the rational matrix function W (z) from samples,

W k = W (zk ), k = 1 , . . . , M,

at arbitrary distinct points in the complex plane was wasintroduced in [9] as Algorithm 2 for identifying stable sys-tems from frequency response data. However the same

algorithm can, without any changes, also be used to iden-tify any rational matrix function. The algorithm is basedon SVD and QR-factorization of certain data matrices,see [9] for a more comprehensive treatment. The approx-imation property of the algorithm is summarized in thefollowing theorem.

Theorem 1 ([9]) Let W (z) be a proper rational matrix of minimal order n . Let W k = W (zk ) + k , k = 1 , . . . , M be perturbed samples of the rational matrix at M distinct points zk (A). Furthermore, let the auxiliary order q > n , M 0 q + n and let W (z) be given by Algorithm 2 in [9]. Then

limk 0

W (z) = W (z)

for all M M 0 .1 The algorithm to be presented can, with obvious changes, be

applied when the rational matrix have complex coefficients

4.2. Basic Approximation AlgorithmThe solution to the approximation problem can be

split into three main steps:

Step 1 Approximation of a rational matrix W (s) RL such that k W k W ( j k )

2 is small. This is donewith Algorithm 2 in [9]. In this rst approximationstep we impose no restrictions on W (s) beside havingno poles on the imaginary axis.

Step 2 First it is checked if W ( j ) + W ( j ) > 0. If nota modication > 0 is introduced

W (s ) = W (s) + I

such that W ( j ) + W ( j ) > 0. To nd a suitable is a convex problem which can be solved by an LMIusing the Kalman-Yakubovich-Popov lemma [15] orby a simple bisection technique checking the eigen-values of the associated Hamiltonian matrix.

A factorization W = Y Z according to Lemma 2 isthen well dened.

Step 3 The obtained factors are converted to some state-space basis suitable for parametrization and the iter-ative parametric optimization of (2) or (3) can beperformed.

4.2.1. Spectral Factorization Since Step 1 inthe algorithm nds a rational approximation to the givendata without imposing any constraints it is most likelythat the obtained approximation do not satisfy W = W ,and consequently Lemma 1 cannot be applied. If so letW := 12 (W + W ) which is Hermitian and Lemma 1 ap-

plies. By this step the order of the factor G is doubled.Prior to the optimization we recommend to reduce the or-der by a balanced truncation. The truncation preserves

the stability of G

. Inverse-stability of the reduced fac-tor can be recovered, if necessary, by a second spectralfactorization.

4.2.2. Positive Real Approximation This ba-sic algorithm can directly be used to the problem givenin equation (3). If modications are necessary in Step 2( > 0) the quality of the approximation obtained by thesubspace method in Step 1 becomes degraded and Step 3is instrumental in order to obtain good results.

4.3. Parametric OptimizationThe success of the parametric optimization of the cri-

terion (2) and (3), i.e. convergence to the global min-

ima, is highly dependent on the quality of the initial es-timate delivered by the subspace algorithm. A second is-sue which is also important is the type of parametrizationused. In this work we have used a recent parametrizationbased on a compact tridiagonal form of the A matrix [10].In our experience the parametric optimization using this

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parametrization has less tendencies to converge to localminima.

5. An ExampleThis example shows an application of LFT gain

scheduling synthesis [12, 8], for a very simple system witha parameter dependency [14]. The system is dened by

x = 0 1 3.25 2.75 0 x +01 u

y = 1 0 x,(5)

where [ 1, 1]. The model describes a pendulum on avertically accelerating platform. The parameter containsboth the gravitational constant and the positive accelera-tion of the platform. The control objectives are to stabilizethe the system, and to reject signicant disturbances atthe input of the plant. The objectives are depicted in Fig-ure 2. The values of the weighting functions from [14] areW p = 0 .5, W n (s ) = 10 s +1s +200 , W d = 1 .33 and W u = 0 .32.The augmented system is of third order. The aim of thedesign is to reduce the L2 -induced gain from w to z as-suming that [ 1, 1].

-u q ?

W u

?z2

g+ ?

W d ?

w2

- M -

- g+ ?

W p ?

w1

- g+ ?

W n ?

w3

-y q

?z1

Figure 2: Augmented system.

5.1. LFT Controllers

We will study some different controllers for this systembased on LFT design techniques. This means that thecontroller is linear and parametrized with respect to asa linear fractional transformation (LFT). We will assumethat is known in real time. First we assume that thereare no bounds on how fast it may vary; it is only boundedto the interval [ 1, 1]. During the design the L2 -inducednorm from w to z is minimized. For a given constant this is equivalent to minimizing the H norm.

Two LFT gain scheduling designs were made: one as-suming to be parametric (real) and the second with dy-namic (complex) . Both controllers are parametrized by and give equivalent performance.

5.2. Constant ParameterSo far we have assumed that the parameter may vary

without bounds on its rate of change. This implies thatthe scaling D must be constant, in order to commute with. We will now assume that is a constant, still bounded

to the interval [ 1, 1]. This allows us to use frequencydependent scalings and multipliers in the design.

We start the D -K iteration by using the gain sched-uled controller in the previous section. The closed loop hastwo uncertainty blocks: one twice repeated scalar blockcorresponding to and one complex full block correspond-ing to the performance criterion ( w to z). The block isrepeated twice since it appears both in the original sys-tem and in the controller. A frequency sweep using 50points is then performed using the mu command in the -Analysis and Synthesis Control Toolbox [1]. The resultingD -scalings for the block is a 2 2 matrix, which can-not be tted with a state-space D -scaling using musynfitsince this command only handles scalar D . Note thatany unitary left factor, U , in D is irrelevant as long asD U UD = D D D mu D mu .

102

101

100

101

102

0

0.5

1

1.5

2

2.5

3

m a g n

i t u

d e

D scalings

Figure 3: D -scalings for the LFT gain scheduled system.The D -scalings are 2 2 frequency dependent matrices.The solid lines show the singular values of the D mu -scalingsfrom the mu command. The dashed lines show the singularvalues of the state-space approximation, D . The dashed-dotted line shows the error between these two scalings asthe maximum singular value of D mu D mu D ( j )D ( j ).

5.3. LFT Gain Scheduling with D -scalingsUsing the D -scalings from the rst iteration, we can

proceed by designing another LFT gain scheduled con-troller. The original system is augmented with the dy-namic scalings as is depicted in Figure 4. The augmentedsystem G has two inputs and two outputs. In addition tothe input and output of th original system G it as extra

input and output going through the scalings, D and D 1

.Thus, the gain scheduled controller, K , has access to via augmented inputs and outputs through the scalingsD and D 1 . However, since the augmented system hasa twice repeated scalar parameter, the controller will alsobe parametrized by a twice repeated block (in addition

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K

G

! !

! !

a a

a a

--

D D 1

G

Figure 4: The augmented gain scheduling problem. Thedashed box, denoted by G , represents the augmented sys-tem, which has feed-through connections between the scal-ings and the controller K .

to the accessible through G ).The static or frozen H performance of the two LFT

gain scheduling controllers, with and without D -scalings,are given in Figure 5. In addition the theoretical lowerlimit is given as the frozen performance of an H con-

troller designed for each value of . As can be seen thisLFT controller is quite close the theoretical limit in theinterval [ 1, 1]. Note also, that since the controlleris parametrized as a linear fractional transformation, it iscontinuous in the parameter . This is not the case if anH controller is computed for each distinct value of ;then continuity cannot be guaranteed.

2 1.5 1 0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

H

n o r m

Static H performance

Optimal

LFT

LFT with D -scalings

Figure 5: Static or frozen H performance of three con-

trollers: an LFT controller (dashed line) without assumingany bounds on the rate of change of , an LFT controller(dash-dotted line) obtained using frequency dependent D -scalings, and a lower bound (solid line) using static Hcontrollers designed at each .

6. ConclusionsA new method for nding matrix-valued real-rational

functions to frequency data is proposed. Specically themethod is intended to be used for D -K iterations in -design. The proposed method can handle matrix valuedscalings, which are required when repeated uncertaintiesare present. The method is illustrated by an example of D -scalings on an LFT gain scheduled controller.

References[1] G. Balas, J. Doyle, K. Glover, A. Packard, and

R. Smith. -Analysis and Synthesis Toolbox for Use with Matlab, Users Guide . The MathWorks, Inc.,1993.

[2] J. C. Doyle. Structured uncertainty in control systemdesign. In IEEE Proceedings of the 24th Conference on Decision and Control , pages 260265, Fort Laud-erdale, Florida, December 1985.

[3] J. C. Doyle, K. Lentz, and A. Packard. Design ex-amples using -synthesis: Space shuttle lateral axisFCS during reentry. In IEEE Proceedings of the 25th Conference on Decision and Control , volume 3, pages22182223, Athens, Greece, December 1986.

[4] 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, pages12271232, Brighton, England, December 1991.

[5] M. Fan, A. Tits, and J. C. Doyle. Robustness inthe presence of mixed parametric uncertainty and un-modeled dynamics. IEEE Transactions on Automatic Control , 36(1):2538, January 1991.

[6] B. Francis. A Course in H Control Theory , vol-ume 88 of Lecture Notes in Control and Information Science . New York: Springer Verlag, 1987.

[7] A. Helmersson. Applications of mixed- synthesisusing the passivity approach. In Proceedings of the 3rd European Control Conference , volume 1, pages165170, Rome, Italy, September 1995.

[8] A. Helmersson. Methods for Robust Gain Schedul-ing . PhD thesis, Linkopings universitet, Link oping,Sweden, 1995.

[9] T. McKelvey, H. Akcay, and L. Ljung. Subspace-based multivariable system identication from fre-quency response data. IEEE Trans. on Automatic Control , 41(7):960979, July 1996.

[10] T. McKelvey and A. Helmersson. State-spaceparametrizations of multivariable linear systems us-ing tridiagonal matrix forms. In Proc. 35th IEEE Conference on Decision and Control , pages 36543659, Kobe, Japan, December 1996.

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[11] T. McKelvey and A. Helmersson. A subspace ap-proach for approximation of rational matrix functionsto sampled data. In Proc. 35th IEEE Conference on Decision and Control , pages 36603661, Kobe, Japan,December 1996.

[12] A. Packard. Gain scheduling via linear frac-tional transformations. Systems & Control Letters ,22(2):7992, February 1994.

[13] A. Packard and J. Doyle. The complex structuredsingular value. Automatica , 29(1):71109, January

1993.[14] A. Packard and Fen Wu. Control of linear fractional

transformations. In IEEE Proceedings of the 32nd Conference on Decision and Control , volume 1, pages10361041, San Antonio, Texas, December 1993.

[15] A. Rantzer. A note on the Kalman-Yacubovich-Popov lemma. In Proceedings of the 3rd European Control Conference , volume 3, part 1, pages 17921795, Rome, Italy, September 1995.

[16] S. Tffner-Clausen, P. Andersen, J. Stoustrup, andH. H. Niemannn. A new approach to -synthesis for

mixed perturbation sets. In Proceedings of the 3rd European Control Conference , volume 1, pages 147152, Rome, Italy, September 1995.

[17] P. Young. Controller design with mixed uncertain-ties. In Proceedings of the American Control Confer-ence , volume 2, pages 23332337, Baltimore, Mary-land, June 1994.

[18] P. Young, M. Newlin, and J. Doyle. analysis withreal parametric uncertainties. In IEEE Proceedings of the 30th Conference on Decision and Control , vol-ume 2, pages 12511256, Brighton, England, Decem-

ber 1991.[19] P. Young, M. Newlin, and J. Doyle. Practical com-

putation of the mixed problem. In Proceedings of the American Control Conference , volume 3, pages21902194, Chicago, Illinois, June 1992.

[20] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control . Prentice Hall, Upper Saddle River,NJ, 1996.

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