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Model-Controller Reduction for Flexible Structures AchievingRobust Performance

Nabil Aouf, Benoit BouletDepartment of Electrical and Computer Engineering and

McGill Centre for Intelligent MachinesMcGill University

3480 University Street, Montréal, Québec, Canada H3A 2A7aouf@cim.mcgill.ca, boulet@cim.mcgill.ca, tel: 514-398-1478 fax: 514-398-4470

Ruxandra BotezDépartement de Génie de la Production Automatisée

Ecole de technologie supérieure1100 Notre-Dame Street West, Montréal, Québec, Canada H3C 1K3

ruxandra@gpa.etsmtl.ca

Abstract: This paper presents a new model-order reduction method applied to a flexiblestructure. Our approach, based on robust closed-loop stability and performance criteria,determines which flexible modes can be truncated from the nominal model whilepreserving closed-loop stability and performance. We use parametric uncertainty toinclude both the uncertainties in the damping ratios and frequencies of the truncatedmodes and the effect of truncation of these modes from the nominal model.

Introduction:

Detailed dynamic modeling of flexible space structures or aircraft typically leads to theinclusion of several flexible modes in the model. Model reduction is then often neededfor the following reasons:(1) It may be easier to analyze the interactions of the most important modes in the

structure on a reduced-order model.(2) A linear time-invariant controller designed using ∞+ or 2+ techniques and based on

a reduced model will itself be of lower order, hence easier to implement,(3) A reduced-order model may be appropriate for quick simulations of different

scenarios.

In this paper, we are mainly focusing on the second reason. Namely, given a flexiblestructure model with N flexible modes, a performance specification, and a limitednumber of modes k N< to be truncated, we want to find the "best" reduced model-controller pair for which the controller generates the closest sensitivity to the nominalsensitivity on the full-order model. The robustness issue is key here not only for thetruncated modes that may cause spillover in closed loop, but also to uncertainty in thesemodes and the retained ones. From the point of view of analysis, it is also desirable toobtain a reduced-order model in modal form matching a reduced controller, rather thanjust a reduced controller, as obtained with standard controller reduction methods.

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Many methods have been developed for order reduction of linear time-invariantcontinuous-time systems. Balanced reduction [5], cost analysis [13], and optimal Hankel-norm approximation [6], are the most popular of these methods, but most of them areused for open-loop reduction. The works of Enns [1], [2] led to what is called weightedbalanced reduction which can be applied to stable plants, taking into account closed-loopstability preservation. Anderson extended this method by finding weights for closed-loopperformance preservation. These methods have been usually applied for controllerreduction [7] rather than plant reduction, which is arguably more complicated because thecontroller is not provided in advance. This "chicken and egg" problem naturally leads tomultiple-step approaches.

Our approach was developed to reduce flexible aircraft models, but in this paper we willmore generally study the truncation of flexible modes from flexible structure models.Low-order (or reduced) controllers are designed based on the best truncated models. Wetake into account robust performance and a "closeness" measure to the ideal closed-loopsensitivity obtained with a full-order controller applied on the nominal model. Thistechnique leads to an algorithm that is simpler than the ones given in [1], [3], [4], andarguably more practical in that it can handle realistic constraints such as closed-loopperformance and uncertainty in the truncated modes.

The fist stage of the algorithm consists of testing all combinations of a fixed number ofmodes that can be truncated, and yields the best candidate combinations for truncationmaximizing the closed-loop robust performance criterion. We use the novel idea thatparametric uncertainty in the candidate flexible modes in the full-order model canrepresent the effect of the truncation of these modes. We use a mixed-µ setup in ourapproach to analyze robust performance. The set of best candidate mode combinations isthen processed a second time to find the "best" combination for truncation. This is doneby designing corresponding (reduced) controllers based on the candidate truncatedmodels, which are then tested for stability of the closed loop with the full-order model.The subset of stabilizing controllers are finally tested by computing the distances in ∞+

of their corresponding closed-loop sensitivities when applied to the full-order model, tothe best possible sensitivity obtained with the optimal full-order controller.

Due to the complete enumeration of all combinations of k modes in the algorithm, it islimited to models with a relatively small number of flexible modes. However, it is notunreasonable to assume that a few days of intense computations would represent arelatively small cost in the overall design of a new commercial aircraft or spacecraft withflexibilities.

Problem Setup

The nominal transfer matrix model of a flexible structure is expressed from its modalstate-space realization as:

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0 0 10 0 0 0 0

0 0

( ) : ( )A B

G s C sI A B DC D

− = = − +

, (1)

where nnRA ×∈0 is in modal form, i.e., 0 { }iA diag A= , 21

21

i i i ii

i i i i

Aζ ω ζ ω

ζ ω ζ ω

− − − − −

= 1, ,i N= ! ,

and N is the number of flexible modes of our model; ii ωζ , are the damping ratio and

the undamped natural frequency of the ith mode; 20

N mB ×∈� , 20

p NC ×∈� , 0p mD ×∈� .

The truncation of the ith mode from the nominal model can be seen as eliminating theeffect of this mode. Because there is no interaction between the modal states in the modalform used, this truncation corresponds to setting to zero the 2 2× matrix iA and the

corresponding rows and columns of the matrices 0B and 0C respectively from the

nominal state-space representation of 0( )G s .

Uncertainty Model:

Parametric uncertainty may be less conservative than other types of uncertainties andmay lead to a more realistic representation, especially parametric uncertainty in theneglected (truncated) modes. In our approach, we treat both this kind of uncertainty andthe truncation of the corresponding modes through the same setup and with the use of asingle real scalar perturbation. We ask for a maximum of performance and robustnessagainst an uncertainty set containing the parameter uncertainty and the effect oftruncation as well. This allows us to determine which are the best candidate combinationsof truncated modes to take into account as a first ordering of the algorithm. A secondordering leads to the best reduced model-controller pair achieving our desired criterion ofcloseness to a desired closed-loop sensitivity in ∞+ .

For a fixed number of flexible modes to be truncated k , we define the set of all possiblecombinations of k flexible modes to be truncated as follows:

{ }1 1: { ,....., } : {0,1},

N

k N i iikα α α α α

== = ∈ =∑� , (2)

where 1=iα to truncate and 0=iα to keep the ith mode. This set contains :Nk

NC

k

=

candidate combinations. For each combination, we define corresponding perturbations ofthe modal state-space matrices representing modal parameter uncertainty and thetruncation effect of these specific modes. First, let { }1 2 2: , , NT diag I Iα α= ! . The

perturbed plant model is then defined as follows:

0 0

0 0

( ) ( )( ) :

( ) ( )A B

C D

A BG s

C D

α αα α

+ ∆ + ∆ = + ∆ + ∆

(3)

where the perturbations of the state-space matrices (see [10]) are defined by

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: , , 1A B

C D

A B

C Dα α

α α

δ δ δ∆ ∆

= ∈ ≤ ∆ ∆ � , (4)

and the truncation matrices , , ,A B C Dα α α α are given by

0 0 0 0, , ,A TA B TB C C T D Dα α α α= = = = . (5)

The real uncertainty parameter δ varies between –1 and 1, covering more than a singleobjective. The correspondence between values of the parameter δ and the objectives inour design is given as follows:

] [ ] ]

1

0

1,0 0,1

δδδ

= − = ∈ − ∪

(6)

By including the matrices , , ,A B C Dα α α α representing the uncertainty and the effect of

the truncation, the augmented plant model to be controlled is given in Figure 1. In fact,these matrices can be lumped in the augmented plant model. The multiplicity of theuncertainty δ represented in Figure 1 can be high. Using a singular value decomposition

of A B

C Dα α

α α

, we can reduce the number of repeated perturbations δ , which leads to a

less conservative uncertainty model:

[ ] 11 2 1 1 1 1

2

0

0 0rT

r

A B VU V U U U V U W

C D Vα α

α α

Σ = Σ = = Σ =

, (7)

Mode truncation

Parametric uncertainty of the truncated modes

Nominal model

Figure 1 Figure 2

0 0

0 0

0

0

0 0 0

0 0 0

A B I

C D I

I

I

1s I−

0

0

I

I

δδ

uy

Figure 3

0 0

0 0

0

0

0 0 0

0 0 0

A B I

C D I

I

I

1s I−

uy

[ ]1UrIδ1W

y

1s I−

u0 0 11

0 0 12

11 12 0

A B U

C D U

W W

rIδ

x� x

z w

A B

C Dα α

α α

Mα

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where max{2 ,2 }0

A Br rank N m N p

Cα α

α

= ≤ + +

and 1 1rW V= Σ . Figures 2 and 3 show,

respectively, how the multiplicity of δ is reduced to r and the matrices [ ]1 11 12W W W= ,

111

12

UU

U

=

are incorporated in the augmented plant.

The reduction in the multiplicity of δ yields a more accurate and less conservativeuncertainty set. The transfer function between y and u is represented by:

1( ) ( , ),L U rG s M s I Iα δ− = � � . Following the standard µ -synthesis scheme, we need to

include the term 1s I− in the dynamics of the system, so we define ( )y u

H sz wα

=

such

that 1( ) ( , )UH s M s Iα α−= � . Furthermore, by reversing the order of the inputs and outputs

of ( )H sα , we obtain the generalized plant model ( )Q sα , as given by Figure 4, mapping

w

u

to z

y

and representing a standard setup for µ -synthesis providing robustness

against a single repeated real perturbation.

Since we are not just interested in robust stability as a selection criteria of the candidatemode combinations for truncation, we introduce the notion of robust performance as aprincipal objective that truncated models have to achieve with a full-order controller.Inputs and outputs of interest (reference signals, disturbances, tracking error, etc.) arethus added to ( )Q sα together with performance weighting functions to obtain the

generalized plant model ( )P sα mapping

w

d

u

to

z

e

y

.

y u

rIδz w

K

Figure 4 Figure 5

y u

rIδz w

K

de

p∆

( )Q sα ( )P sα

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Figure 5 shows the −µ synthesis setup for robust performance. nd nep

×∆ ∈� is a fictitious

uncertainty included for performance, linking the exogenous inputs d to the outputs to becontrolled e. The uncertainty structure is defined as follows:

0: : ,

0d ep n n

p δδ

× ∆ Γ = ∆ = ∆ ∈ ∈

� � . (8)

We use mixed-µ theory [12], to synthesize a controller achieving the best robustperformance index. Since our technique uses only real scalar uncertainty parameter δ ,with a reduced multiplicity, and because we add a complex fictitious performanceuncertainty p∆ , it should be easier to find the µΓ bound than for the case of a single real

perturbation [15]. Note that if a given combination α leads to a robust performance index

{ }( ), ( ) 1L P s K sαµΓ < � , then this combination of modes can be truncated from the

model and the closed loop would still achieve the desired performance specification.

Reduction technique:

Based on the setup proposed in Figure 5, the most important criterion taken into accountin the first stage of our technique is the achieved robust performance level. Thatobviously means that we use the closed-loop system, with all possible combinations ofmodes to be truncated, rather than the open-loop system model.

Model reduction based on open-loop models, and for various norms, features the usefuladditivity property, i.e., each mode adds an independent contribution to a given criterion(say closeness to the full-order model). This property greatly facilitates modal truncationin open loop: the contribution of each mode to the criterion is computed separately, andthe best combination of k modes to be truncated is just the k modes with the smallestcontribution to the criterion. On the other hand, our closed-loop based algorithm must testall combination of k modes. However, it includes the two essential notions of robuststability and performance that are critical for control applications. We based our methodon an two-stage ordering. The first ordering uses the robust performance criterion and thesecond one checks for closed-loop stability and computes performance in terms ofcloseness to a desired sensitivity in ∞+ . Our reduction technique not only finds a goodreduced-order model but also its associated best reduced controller, providing a goodtradeoff between stability and performance of the reduced controller on the nominal full-order model.

Algorithm:

Our model reduction algorithm can be formulated as follows:

Initialization:- Let 0( )G s be the nominal model and 0 ( )K s be an ∞+ -optimal controller for it.

- Let N be the number of flexible modes in the nominal model.

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- Fix k, the number of flexible modes to be truncated.- Fix the minimum robust performance required minP .

- Generate the set k� of NkC candidate combinations of k truncated modes.

- φ=L is the set of candidate combinations in k� that meet the robust

performance criterion.

Ordering 1:For 1, , N

kj C= !

- Design a full-order controller ( )jK s using a µ -synthesis technique with respect

to the structured uncertainty Γ for mode combination jkα ∈� .

- Compute { }( ), ( )jj L jP P s K sαµΓ = � , the robust performance index with mode

combination jα .- If minPPj ≤ then jL L α= ∪ and there exists a full-order 0 ( )K s that reaches the

minimum robust performance specification minP for this thj candidatecombination.

Ordering 2:- Let pn be the number of mode combinations retained from Ordering 1 (number of

elements in L) For 1, , pm n= !

- Reduce the nominal model 0( )G s to ( )rmG s as given by mα .

- Design a reduced controller rmK for the reduced model rmG and compute the

closed loop transfer matrix ( , ).L rm rmG K�

- If ( , )L rm rmG K� is unstable, then this mth candidate is automatically rejected.

- Else:Calculate 0 0( , ) ( , )m L L rm rmP G K G K ∞= −� � and store it in a list P.

End End

The best reduced model-controller pair ,r rG K is the pair associated with

min{ : }m mP P P∈ .

Numerical Example:

To illustrate our method we chose a flexible system taken from [14], consisting of threemasses 10,5,11 321 === mmm , stiffness 55,50,10 3241 ==== kkkk , and damping

ii kd 01.0= , .4,3,2,1=i The input u is applied such that ufufuf 5,2, 321 −=== where

, 1,2,3if i = are the forces applied on each mass respectively. The output is

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1 1 2 32 2 3y q q q= − + , where , 1,2,3iq i = are the mass displacements. A disturbance d, is

added to the plant output 1y to get the system output 1y y d= + . We represented thissystem in the state-space modal form such that the nominal damping ratios andfrequencies respectively are: 1 1 2 20.025, 5.12, 0.012, 2.43,ζ ω ζ ω= = = =

3 30.0044, 0.87ζ ω= = .

For our model, we need to find which flexible modes can be truncated without giving uptoo much performance of the nominal model. As presented previously, the parametricuncertainty of the candidate mode to be neglected will cover also the effect of thetruncation of this candidate. The performance and the constraints on the controllerspecified for the nominal model are given by the weighting functions

3( )

60.5 0.03p

sW s

s

+=+

and 1.0=uW , respectively. ( )pW s specifies that the closed-loop

sensitivity should be smaller than 0.01. The sensitivity consists in this case of the transferfunction matrix ( )dyT s from the disturbance d to the output of the system y. The

parametric uncertainty δ , for a “combination” containing a single mode to be truncated,has multiplicity 3. As we consider combinations with more than a single mode themultiplicity of the perturbation δ increases. Note that for each additional mode to betruncated in a combination, the uncertainty set will be augmented by two replications ofthe same perturbation δ . Added to this parametric uncertainty is the robust performance(fictitious) uncertainty p∆ , which is complex and normalized by the weighting function

( )pW s .

A µ-design is performed for such a model to analyse the robust performance obtainedunder the previous considerations. A well-known solution to find the µ-bounds isprovided by the D-K iteration algorithm [11]. One criticism about this algorithm is that itoften cannot deal with high-order systems, or even with an increasing number ofuncertainties which in general reflects realistic constraints. We noticed that when weapplied our proposed reduction algorithm for the mixed µ design, a large number ofiterations were required to approach the actual value of µ, which led to an increase in thenumber of states of the augmented model. This made the computation of a controller verydifficult. For this reason we investigated alternative algorithms that avoid the drawbackof the D-K iteration algorithm, and approach ( [ ( ), ( )])jl jF P s K sαµΓ with lower-order

Figure 6

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augmented plant models. The approach used is a µ-synthesis based on the iterativecomputation of optimal 2+ controllers [9].

Our proposed algorithm gives the designer the freedom to specify from the setup thenumber of modes to truncate and the minimal robust performance index during the firstordering. For this example, we wanted to find the best reduced models by truncating oneflexible mode, and two flexible modes respectively. Table 1 shows the results of ouralgorithm applied to the three-mass flexible system model when we ask to truncate oneflexible mode. The first ordering takes into account the single modes truncations thatachieve at least a robust performance index of 0.8. The best choice was the first“combination” with 637.01 =p . This led in the second ordering, with the correspondingreduced controller, to closed loop stability and an index of closeness to the nominalclosed loop of 41 0.026 10h −

∞ = × . If we relax the robust performance specification by

fixing the minimum index to 1.0012, we obtain in the first ordering two possiblecombination candidates (combination 1 and combination 2) with their achieved robustperformance indices 637.01 =p and 0011.12 =p , respectively. In this case, the bestcombination obtained from the second ordering is combination 2 rather than combination1, which was the best one for the tighter robust performance specification. This makesense because in the first stage of the algorithm, the emphasis is on robustness andperformance, but in the second ordering, we are asking for maximal closeness to thenominal closed-loop system by the index

∞hi . Hence, the designer has the freedom to

make a trade-off between robust performance and approaching the behaviour of thenominal full-order closed loop system without uncertainty. In general, especially whenone has to consider a larger number of flexible modes, a good choice would appear to beplacing a strict requirement on robust performance in the first stage, and take thecombination with the best index in the second stage.

Table 2 shows the results when we want to truncate two flexible modes from the nominalmodel. For this case we asked for less robust performance than in the first case. If wewould have specified the same robust performance, we would not have been able to findany combination of two truncated flexible modes achieving our objectives, which wouldimply that no reduction to one flexible mode is acceptable. From Table 2 we remark thatit was also required to make a certain trade-off between robust performance and nominalbehaviour. The best indices

∞hi found in this case are greater than in Table 1, which

confirmed the difficulty to get a good reduced model/controller pair when the number oftruncated flexible modes increases.

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Table 1

First ordering : Robust performanceSecond ordering: stability and nominal performance

0 0 0[ ( ), ( )] [ ( ), ( )]l rm lF P s K s F P s K s∞

−

Combinations Combinations

304.13 =p 43 0.5118 10h −∞ = × 5.1min =p

2min =p 87.12 =p 304.13 =p 42 0.5104 10h −∞ = ×

Table 2

First ordering : Robust performanceSecond ordering: stability and nominal performance

0 0 0[ ( ), ( )] [ ( ), ( )]l rm lF P s K s F P s K s∞

−

Combinations Combinations

637.01 =p 41 0.026 10h −∞ = × 5.1min =p

2min =p 637.01 =p 0011.12 =p 42 0.0195 10h −∞ = ×

Conclusion:

We introduced a new approach for model-controller reduction. This approach takes intoaccount the closed-loop stability and performance preservation as criteria for truncationof the flexible modes. Using the proposed algorithm, one can find the best reducedmodel, from a list of possible candidates, achieving the desired robust performance inclosed loop. Then, through a second ordering, the corresponding best (with respect tostability and sensitivity) reduced controller is selected. The method proposed has theability to represent the effect of mode truncation by a real parameter uncertaintyrepresenting also the uncertainties in the neglected modes, an important aspect to takeinto account. This is not the case for many papers published in this field of research. Weused an efficient method to find the µ bound, the index of the first ordering, which wasdifficult to compute with classical tools such as D G K− − iteration. The results obtainedin the numerical example show that our method offers a tradeoff that might correspondbetter to the real needs of the control engineer in trying to find a reduced model-controller pair.

Bibliography:

[1]- D. Enns. Model reduction for control system design. PhD thesis, Stanford University,1984.[2]- D. Enns. Model reduction with balanced realizations: an error bound and frequencyweighted generalization. In proceedings of the 23rd conference on Decision and Control,Las Vegas, Dec 1984. IEEE p127-132.

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[3]- Madelaine, B and Alazard, D. Flexible structure model construction for controlsystem. In Proceedings of the Conference on Guidance Navigation and Control. AIAA,Boston 1998 p 1165-1175.[4]- Madelaine, B and J.-P. Chretien. Weighted balanced model reduction of a flexiblestructure. In Proceedings of the American Control Conference, Philadelphia. June 1998.[5]- B.C. Moore. Principal component analysis in linear systems: Controllability,Observability and model reduction. IEEE Transactions on Automatic Control, 26(1), Feb1981.[6]- K. Glover. All optimal Hankel-norm approximations of linear multivariable systemsand theirL∞ bounds. IEEE Transactions on Automatic Control, 39(6): 1115-1193, 1984.[7]- B.D.O. Anderson and Y. Liu. Controller reduction: Concept and approaches. IEEETransactions on Automatic Control, 34(8), Aug 1989.[8]- Yang, C.-D and Chang, C.-Y., µ -Synthesis Using Linear Quadratic GaussianControllers. Journal of Guidance, Control and Dynamics, Vol 19 No 4. 1996 pp 886-892.[9]- Yang, C.-D , Chang, C.-Y and Y.-P. Sun., Mixed µ -Synthesis via 2H -Based Loop-Shaping Iteration. Journal of Guidance, Control and Dynamics, Vol 21 No 1, Engineeringnotes. 1997[10]- B.Morton and R.McAfoos, A Mu-test for real parameter variations. In Proceedingsof the American Control Conference, pp 135-138, 1985.[11]- Balas, G.J., Doyle, J.C., Glover, K., Packard, A., Smith, R., 1995 µ -Analysis andsynthesis toolbox MUSYN Inc, and the MathWorks, Inc.[12]- Young, P.M and J.C. Doyle. Computation of µ with real and complexuncertainties. In Proceedings of the 29th IEEE Conference on Decision and Control, pp1230-1235, 1990.[13]- R.E. Skelton. Cost decomposition of linear systems with application to modelreduction. In International Journal of Control, 32 (6): 1031-1035, 1980.[14]- Gawronski, W, Balanced Control of Flexible Structures. Lecture Notes in Controland Information Sciences 211 Springer-Verlag London Limited 1996.[15]- Packard, A and Pandey.P. Continuity of the real/complex structured singular valueIEEE Transactions on Automatic Control, Vol. 38 no 3 pp415-428, March 1993.