chapter 1 model reductionand real-time control for...

Chapter 1

MODEL REDUCTION AND REAL-TIME CONTROLFOR DYNAMIC DATA DRIVEN SYSTEMS

A. Antoulas1, D. Sorensen2, K. A. Gallivan3, P. Van Dooren4, A. Grama5, C.Hoffmann6, and A. Sameh7

1Department of Electrical and Computer Engineering, Rice University, Houston, TX

[email protected]

2Department of Computational and Applied Mathematics, Rice University, Houston, TX

[email protected]

3 School of Computational Science, and Information Technology, Florida State University, Tal-lahassee, FL

[email protected]

4 Department of Mathematical Engineering, Catholic University of Louvain, Louvain-la-Neuve,BELGIUM

[email protected]

5Department of Computer Sciences, Purdue University, W. Lafayette, IN

[email protected]


[email protected]


[email protected]

Abstract Simulation and control are two critical elements of Dynamic Data-Driven Ap-plication Systems (DDDAS). Simulation of dynamical systems such as weatherphenomena, when augmented with real-time data, can yield precise forecasts.In other applications such as structural control, the presence of real-time datarelating to system state can enable robust active control. In each case, there is

2 DDDAS

an ever increasing need for improved accuracy, which leads to models of highercomplexity. The basic motivation for system approximation is the need, in manyinstances, for a simplified model of a dynamical system, which captures themain features of the original complex model. This need arises from limitedcomputational capability, accuracy of measured data, and available storage ca-pacity. The simplified model may then be used in place of the original complexmodel, either forsimulation and prediction, or active control. As sensor net-works and embedded processors proliferate our environment, technologies forsuch approximations and real-time control emerge as the next major technicalchallenge. This paper outlines the state of the art and outstanding challengesin the development of efficient and robust methods for producing reduced ordermodels of large state-space systems.

Keywords: Model Reduction, Real-time Control, Dynamical Systems, Linear Time Invari-ant Systems, Direct Numerical Simulations, Parallel Algorithms.

1. Introduction

Many physical processes in science and engineering are modeled accu-rately using finite dimensional dynamical systems. Examples include weathersimulation, molecular dynamic simulations (e.g., modeling of bio-moleculesand their identification), structural dynamics (e.g., flex models of the inter-national space station, and structural response of high-rise buildings to windand earthquakes), electronic circuit simulation, semiconductor device manu-facturing (e.g., chemical vapor deposition reactors), and simulation and con-trol of micro-electro-mechanical (MEMS) devices. An important subclass ofthese processes can be effectively monitored to gather data in support of sim-ulation, diagnosis, prognosis, prediction, and control. This class of dynamicdata-driven application systems (DDDAS) pose challenging problems rangingfrom data gathering, assimilation, effective incorporation into simulations, andreal-time control.

In the vast majority of applications where control can be affected, the origi-nal system is augmented with a second dynamical system called a controller. Ingeneral, the controller has the same complexity as the system to be controlled.Since in large-scale applications of interest, one often aims at real-time control;reduced complexity controllers are required. One possible solution to buildinglow-complexity controllers is to design such controllers based on reduced ordermodels of the original dynamical systems. In yet other systems in which real-time prognostics and prediction are required, constraints on compute power,memory, communication bandwidth, and available data might necessitate theuse of such reduced-order models. This paper outlines the state-of-the-art inmodel reduction, discusses outstanding challenges in the context of a varietyof applications, and presents possible solution strategies for addressing thesechallenges.

Model Reduction and Real-Time Control for Dynamic Data Driven Systems 3

Model reduction seeks to replace a large-scale system of differential ordifference equations by a system of substantially lower dimensions that hasnearly the same response characteristics. Two main themes can be identifiedamong several methodologies: (a)balancing based methods, and (b)momentmatching methods. Balanced approximation methods are built upon a familyof ideas with close connection to singular value decomposition. These meth-ods preserve stability and allow for global error bounds. Available algorithmsfor these methods, however, are not suitable for large-scale problems sincethey have been developed mainly for dense matrix computations. Momentmatching methods are based primarily on Pade-like approximations, which,for large-scale problems, have led naturally to the use of Krylov and rationalKrylov subspace projection methods. While these moment matching schemesenjoy greater efficiency for large-scale problems, maintaining stability in thereduced order model cannot be guaranteed. Consequently, their use can beproblematic at times. Moreover, no a priori global error bounds exist for mo-ment matching schemes.

A current research trend aims at combining these two approaches by deriv-ing iterative methods that incorporate the desirable properties of both of theseclasses. This chapter addresses several important unresolved issues in modelreduction of large-scale Linear Time-Invariant (LTI) systems, and extendingthem to a new class of problems that require adaptive models. It also addresseslarge-scale structured problems that are either time-varying, or which requireiterative updating of the initial reduced models to obtain better approximationproperties. All of these are key distinguishing characteristics of a large classof dynamic data-driven systems.

2. Motivating Data Driven Control Applications

The problems addressed in this paper focus on reduced models for dynamicdata-driven systems, which are characterized by model adaptivity and need formaintaining or exploiting a specific type of structure. This can be either dueto time variations in the model itself, or due to the fact that the model to beapproximated varies at each iterative step of some design loop.

Large-scale mechanical systems:The field of sensor networks and dense in-strumentation has seen intense research and development activity in the re-cent past (Hill et al., 2000; Kahn et al., 1999; Li et al., 2001; Goodman,2002; Bulusu et al., 2000; Bhargavan et al., 1994; Woo and Culler, 2001; Yeet al., 2002; Gutierrez et al., 2001; Haas et al., 1999; Chang and Tassiulas,2000; Royer and Toh, 1999; Sohrabi et al., 2000; Haas, 2000; Cerpa and Estrin,2002). These activities have focused on the network layer, communications’abstractions, data assimilation, power consumption, and related enabling com-putational infrastructure. As these technologies mature, the focus shifts to the

4 DDDAS

use of these sensor networks and associated computational elements to ana-lyze and effectively control the underlying systems. While problems in controlhave been studied extensively over the past decades, the use ofsensor-basedmicro-gridsas platforms for high-fidelity real-time control of complex struc-tures poses significant new challenges. The complexity of underlying systemscoupled with the fine granularity of actuation mechanisms result in extremelylarge numbers of degrees of freedom. The distributed computing micro-gridson which the control algorithms must execute require high degrees of con-currency and low communication overheads. Real-time constraints on controlnecessitate solutions to complex problems, often in few hundreds of millisec-onds or less. Moreover, the error-prone nature of many of the underlying sens-ing networks require robust control mechanisms. Current generation of controlmechanisms tend to be largely passive – relying on a static model and a passivecontroller, designed to damp dominant perturbations to the systems (e.g., dom-inant frequencies of vibrations of buildings). The promise of DDDAS controlis in affecting real-time active control.

Consider simulating a structure that is discretely modeled by a large scalefinite element formulation such as a long-span bridge, a tall building, or evena car windshield that is subject to various external forces. In tall buildings,for example, these forces could be the result of strong ground motion or wind.Typically, the model is specified by second-order systems of equations of theform:

M(ω, t)x(t) + D(ω, t)x(t) + K(ω, t)x(t) = f(t) + Bu(t),y(t) = Cx(t). (1.1)

Here,M(ω, t) is the mass matrix,D(ω, t) is the damping matrix, andK(ω, t)is the stiffness matrix of the system. All three matrices are assumed to befrequency-dependent and possibly time-dependent. For large-scale structures,the state vectorx(t) is of orderN , which can reach in the tens of millions. Theforcing functionf(t) represents wind and earthquake effects and is of the samedimension. We assume that the structure under consideration is densely instru-mented by networked sensing and actuator elements. Further, assuming that asensor-actuator complex (SAC) is monolithically integrated with a strut systemcontaining controllable dampers that can change the stiffness characteristics ofthe structure in milliseconds, the control functionu(t), anm-dimensional vec-tor, represents the action that can be affected through the smart strut system.Finally, thep-dimensional vectory(t) represents the signals collected via thesensor network. The dimensionN of the state vectorx(t), is much larger thanm andp. The objective of the model reduction problem is to produce a modelof the structure that possesses the “essential” properties of the full-order model.Such a reduced order model can then be used for the design of a reduced-ordercontroller to affect real-time control. The reduced-order model may be used


to predict the onset of failure, and the reduced-order controller can be used forreal-time control of the structure so as to mitigate failure.

There are several preliminary problems that must be solved before we re-alize an appropriate reduced order model. These include: (i) computing theoptimal location of the sensors and actuators, where we could start from anominal model withM, D, andK constant, (ii) improve the dynamical modelonce the sensor/actuator placements are determined, using validated simula-tions (via physical laboratory models of the structure) or via measurementsperformed on the actual structure, if possible, (iii) design a closed-loop con-troller that is derived from the reduced-order model and, finally, (iv) adapt thereduced order model to the closed-loop configuration of the system (this willchange its dynamics and therefore, also its reduced order model). In each ofthese steps it is useful to have simplified (i.e., reduced-order) models on whichthe most time consuming tasks can be performed in reasonable time.

Physical verification of VLSI circuits: Verification of VLSI circuits poseschallenges from the computational electromagnetic point of view. TypicalVLSI circuits consist of a chip measuring roughly 1cm× 1cm; the active de-vices, used mainly for signal amplification and switching, are found inside thefirst few micrometers of silicon with a complicated pattern of interconnectionson top of them. The scale of the devices is of the order0.1µm and their numberis several million (more than 50 million in the Pentium IV chip). As a compar-ison we note that the Intel 4004 microprocessor, which was released in 1970,had feature size of10µm, 2300 components, and was operating at 64kHz; theIntel Pentium IV Processor, released in 2001, has feature size of0.18µm, 42million components, operates at over 2 GHz, and the length of interconnect isabout 2 km.

The interconnections, which form an important component of the chip forour considerations, are distributed over several layers, and their length is of theorder of kilometers. Given the operating frequency, which is in the gigahertzrange, these interconnections exhibit parasitic resistive, capacitive, and induc-tive behavior. In the current sub-micron technology, these parasitic effectstend to dominate the overall performance and therefore must be accounted for.Modeling of these phenomena leads to very high dimensional systems that arelinear and passive (since they are composed of RLC elements). A linear sys-tem is passive (i.e., dissipates energy) if its transfer function is positive real. Inorder to expose the faults in thechip design phase, aphysical verification stepmust be performed. The findings of this step are used to modify or fine-tunethe design process.

During chip verification one must solve Maxwell’s equation for very com-plicated geometric shapes. One way to achieve this is via the PEEC (PartialElement Equivalent Circuit) methodology, which amounts to a discretizationof Maxwell’s equations over the given geometry of the chip. The complexity

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of the resulting interconnect models typically involve105 to 106 degrees offreedom. Reduced order modeling aims at producing models of order as lowas possible which nevertheless accurately predict chip behavior. Consequently,model reduction methods with additionalstability andpassivityconstraints areimperative for this set of problems.

3. Linear Algebraic Basis for Model Reduction

Projection methods have provided effective model reduction algorithms forconstructing reduced-order models in the large scale setting. Given a continuous-time state space system:

x(t) = Ax(t) + Bu(t), y(t) = Cx(t), (1.2)

with input u(t) ∈ <m, statex(t) ∈ <N and outputy(t) ∈ <p, one defines aprojectedmodel of reduced ordern� N for a reduced statex(t) ∈ <n :

˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t), (1.3)

whereIn = WTV, A = WTAV, B = WT B, C = CV. (1.4)

The two main approaches to projection methods differ in basic characteris-tics such as the metric used to define the “closeness” of the projected and orig-inal models, and the properties of the original model preserved in the projectedmodel. Recently, however, a much deeper understanding has been realizedfor linear time invariant (LTI) systems of the relationships between the twoapproaches and the tradeoffs involving capability, complexity, and efficiencyof implementation. We have also gained better understanding of the requiredhigh-performance numerical linear algebra infrastructure and hybrid methodsthat combine the strengths of the two approaches.

Balanced approximation methods: The widely used balanced truncationtechnique constructs the “dominant” spacesW andV such that

PW = VΣ+, QV = WΣ+, WTV = In (1.5)

whereΣ+ is symmetric, positive definite, and usually diagonal, andP, Qare the controllability and observability gramians, respectively. It follows thatPQV = VΣ2

+ andQPW = WΣ2+ and henceΣ2

+ should contain the largesteigenvalues ofPQ or QP. The gramians are non-singular if and only if thesystem{A,B,C} is controllable and observable. However, the eigenspacesof (1.5) essentially yield a projected system that keeps those states that are themost observable and controllable simultaneously. This is related to the fact thatfor a given statex, xTP−1x can be viewed as the input energy needed to steerthe system from the zero state tox, andxTQx can be viewed as the energy of


the output resulting from the initial statex (see (Zhou et al., 1995) for a morerigorous discussion).

In the discrete-time case the gramians play the same role, but can be de-finedQ = OTO andP = CCT , whereC =

[B AB A2B · · · ] is the

controllability matrix andOT =[CT (CA)T (CA2)T · · · ] is the ob-

servability matrix. The so-called Hankel map relating past inputs to futureoutput factorizes asH = OC and its singular values squared are the nonzeroeigenvalues ofQP or PQ. Thus, dismissing small Hankel singular valuesyields a good approximation of the Hankel map. Finally, bounds for the errorE(s) = G(s)−G(s) (in an appropriate norm) between both transfer functionsG(s) = C(sIN −A)−1B andG(s) = C(sIn− A)−1B can also be obtainedfrom the Hankel singular values that were dismissed (Zhou et al., 1995). Theseresults all hold for continuous time systems as well with an appropriate defini-tion forH.

Efficient algorithms for the large-scale continuous time case have been de-rived by exploiting the fact that the gramians can be obtained from the Lya-punov equations

AP + PAT + BBT = 0, ATQ+QA + CTC = 0.

and in the discrete-time case from the solution of the Stein equations :

APAT − P + BBT = 0, ATQA−Q+ CTC = 0.

For example, in (Zhou, 2002; Zhou and Sorensen, 2002) an iteration calledAISIAD was developed. The algorithm alternates between two coupled Sylvesterequations that are approximations to the projected equations

A(PW) + (PW)HT + B(BT W) = PFw,

AT (QV) + (QV)H + CT (CV) = QFv,

whereFw = (I − WVT )ATW and Fv = (I − VWT )AV. Given acurrent guessW,V, the algorithm proceeds by determiningZ by solvingAZ+ZHT +B(BT W) = 0, whereH = WTAV andV← V+ is obtainedfrom Z. Y is then determined by solvingATY + YH+ + CT (CV) = 0whereH+ = WTAV+, andW ← W+ is obtained fromY to maintainrelations (1.5).

The AISIAD iteration is effective, but no convergence results are known.These results may be potentially derived from the fact that one can show(Sameh and Tong, 2000; Sameh and Wisniewski, 1982) thatV andW willspan the dominant invariant subspace of thepositive definite generalized eigen-value problems(λP−1 − Q) and(λQ−1 − P) if and only if they satisfy therespective “Tracemin” conditions:

min trace VTP−1V, s.t. VTQV = In,

min trace WTQ−1W, s.t. WTPW = In.

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One possible approach is to recast AISIAD in terms of the Tracemin algo-rithm developed by us (Sameh and Tong, 2000; Sameh and Wisniewski, 1982),working implicitly on the dominant subspace of the productPQ through the al-ternate application of the two coupled Sylvester equations given above. Sincethe Tracemin framework accommodates inexact matrix-vector products andsolutions to the saddle-point problems that arise in the iteration, there shouldbe a means to apply the convergence theory for Tracemin to this modifiedAISIAD iteration.

Interpolation-based methods:Interpolation-based methods are based on thenotion that closeness of the projected and original models should be measuredin terms of interpolation conditions in important portions of the frequency do-main known to the application scientist or engineer. As a result, application-specific insight could be used, and is often required, to guide interpolationpoint placement. For these methods, the projected model approximates theoriginal model at specified pointssi, 1 ≤ i ≤ k, to specified ordersmi,1 ≤ i ≤ k, i.e., G(si) − G(si) = O ((s− si)mi). This is equivalent tosaying that the firstmi coefficients of the Taylor expansions or moments ofG(s) andG(s) aboutsi match, i.e.,C(siIn − A)−kB = C(siIN −A)−kBfor 1 ≤ k ≤ mi andn� N .

Extensive studies have been conducted, and numerous algorithms developedin this area (Antoulas et al., 2001; Antoulas and Sorensen, 2001; Feldman andFreund, 1995; Freund, 1999; Grimme, 1997; Gallivan et al., 2002b; Vanden-dorpe and Dooren, 2003; Jaimoukha and Kasenally, 1997; Kamon et al., 2000).Efficient algorithms, including the Rational Krylov family, that incrementallyproduce the projection matricesW and V are based on the fact that mul-tipoint moment matching is guaranteed for generalized state space multiple-input-multiple-output models if theIm(V) includes the union of the KrylovspacesKmi

((siE−A)−1, (siE−A)−1B

)andIm(W) includes the union

of the Krylov spacesKmi

((siE−A)−T , (siE−A)−TCT

), where thesis

are such that the inverses all exist.For single-input-single-output (SISO) systems, we have recently shown that

multipoint moment matching is completely general. Specifically, any strictlyproper SISO transfer function of McMillan degreen can be produced fromevery strictly proper SISO transfer function of McMillan degreeN > n via asuitably constructed projection. The flexibility that results from this generalityis both the strength and weakness of the interpolation-based approaches. Whilelocal error near the interpolation points is, by definition, small and propertiesof the projected model guaranteed by particular choices of interpolation points,e.g., passivity, the global error can be arbitrarily large. It is, therefore, crucialthat heuristics be developed that can alert one to error growth outside the neigh-borhoods of the interpolation points. Similar concerns arise, for example, inhybrid approaches that exploit the interpolation-based sufficient condition for


passivity of the projected system (see passivity section below). The passivity-producing points may be located in regions of the complex plane that are ofno interest from the point of approximation error. The inclusion form of theKrylov space moment matching conditions above guarantee that we can addnew interpolation points to reduce error appropriately while maintaining inter-polation at the passivity-producing points. However, the new points may causea loss of passivity. There is, therefore, a need to explore heuristics for mon-itoring deviation from passivity in the projected system as new interpolationpoints are added.

For MIMO systems, the generality property does not hold. We have, there-fore, begun exploring the use of tangential interpolation for such systems (Gal-livan et al., 2002a) in a manner that is consistent with multipoint momentmatching for SISO systems. Recently, we have shown that the tangential in-terpolation conditions relatingG(s) andG(s) guarantee thatG(s) is either aminimal unique multipoint moment matching projected model or a combina-tion of multipoint moment matching projection and modal truncation. We havealso made some progress on developing algorithms with efficiency similar tothat of the Rational Krylov family of methods for multipoint moment matchingand continue to investigate their adaptation and use on the structured problemsdescribed later in this chapter.

Sylvester equations:Sylvester equations have emerged as a key link betweeninterpolation-based and balanced approximation-based approaches. On the onehand, interpolation theory can provide a foundation for establishing error esti-mates and convergence properties, while on the other hand, solution of certainSylvester equations through invariant subspaces can provide efficient and reli-able computational schemes. This simple but deep connection establishes thesolution of Sylvester equations as a fundamental tool that arises directly orimplicitly in many algorithms.

It can be shown that, for SISO systems and certain MIMO systems, momentmatching via one- or two-sided projections can be stated in terms of Sylvesterequations (Gallivan et al., 2002a). For example, given mild conditions onW,the matrixV that defines a multipoint moment matching projection(A, B)of a generalized state space model(E,A,B) satisfies the Sylvester equationAV+EVAT +BBT = 0, where the interpolation points are the eigenvaluesof A. This offers an alternative family of algorithms to the Rational Krylovmethods and their incremental nature as well as potential for hybrid methoddevelopment. There is also evidence that Sylvester equations can form thebasis of efficient algorithms for tangential interpolation of MIMO systems.

A typical subproblem arising in many of these algorithms is to derive thesolutions to a sequence of closely related Sylvester equations involving theoriginal large sparse matrixA and a very small matrixH as coefficient matri-ces. Efficient iterative methods may be derived due to the equivalence:

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AZ + ZH + M = 0 iff Z = XY−1,

where [A M0 −H

] [XY

]=

[XY

]R

with σ(R) = σ(−H). The eigenvalues ofR are, in fact, interpolation points.A major advantage of this formulation is that the invariant subspace problem isoften ideally suited to the capabilities of existing modern large scale eigenvaluesoftware (e.g. ARPACK, (Lehoucq et al., 1998)).

Numerical linear algebra infrastructure: The discussion above identifiesthree key areas of infrastructure development for projection-based reductionof LTI models and its generalization to the more difficult structured prob-lems. These are: large-scale invariant subspace computation using Arnoldi-like methods, trace minimization-based large-scale eigenvalue methods, andefficient methods for solving several related, typically by a shift, large sparselinear systems with one or more right-hand side vectors.

The Implicitly Restarted Arnoldi (IRA) method is a synthesis of the im-plicitly shifted QR iteration (for dense problems) with the Arnoldi methods(for large sparse problems). It has resulted in very effective software calledARPACK (Lehoucq et al., 1998; Sorensen, 1992), which is useful for com-puting a selected subset of the eigenvalues of a large matrix. The techniquesused in IRA are closely related to those of model reduction. Thus, any parallelenhancements of IRA are generally applicable to model reduction. A paral-lel version called PARPACK has been developed using a very straightforwardparallel decomposition. This software has been used to solve nonsymmetriceigenvalue problems related to linear stability analysis (in fluid flow) for prob-lems on the order of107 variables. The open research problem here relates tothe development of parallel preconditioners for accelerating the convergenceof IRA. We have been developing such a scheme that effectively does an ap-proximate shift-invert spectral transformation without factoring any matrices.Consequently, this technique may lead to a highly parallel algorithm.

The TRACEMIN algorithm was developed in 1981 (Sameh and Wisniewski,1982) for obtaining the smallest eigenpairs of large sparse generalized eigen-value problemsAx = λBx, whereA andB are symmetric withB positivedefinite. The algorithm is extremely robust. With an effective shifting strategy,or acceleration via Chebyshev iterations, the algorithm compares favorablyto varieties of the Lanczos algorithms, and the related Jacobi-Davidson algo-rithm adapted for symmetric problems. One of the main computational ker-nels needed for efficient implementation is a fast saddle-point problem solver.Solving saddle-point problems has been an area of active research (Sarinand Sameh, 2003; Sameh and Sarin, 2002; Golub et al., 2001; Sameh andSarin, 1999; Tong and Sameh, l998; Sarin and Sameh, 1998). Moreover,


the algorithm is ideally suited for parallelization (Sameh and Wisniewski,1982; Sameh and Tong, 2000). It should be noted that the algorithm has beenused effectively for computing few of the smallest singular triplets of very largesparse matrices, (Berry et al., 1994; Berry et al., 2003).

4. Model Reduction for Structured Systems

In this section, we define three classes of model reduction problems, whichall result in first-order state space formulations with structured matrices.

4.1 Model Reduction of Weighted and Closed-LoopSystems

We are often confronted with model reduction problems of systems com-posed of interconnected parts. The model reduction task consists, in this case,of reducing onlysomebut not all, of the subsystems. The quality of approx-imation is then judged by how close the original overall system is to the oneobtained by replacing the designated subsystems with less complex versions.We note two important problems in this category.

Frequency weighted model reduction. Given a plantG(s) and weightingmatricesWl(s) andWr(s), which may reflect a preference in the frequencycontent of the signals applied to the system, or measured. We seek a plantG(s) of lower complexity thanG(s) such that the error

‖Wl(s)[G(s)− G(s)]Wr(s)‖ (1.6)

for an appropriate norm, is kept small or even minimized. Let

G =[

A BC 0

], Wl =

[Fl Gl

Hl Jl

], Wr =

[Fr Gr

Hr Jr

],

be the system quadruples, then a realization(At,Bt,Ct) for the compoundsystemWl(s)G(s)Wr(s) is given by

[At Bt

Ct 0

]=

Fl GlC 0 00 A BHr BJr

0 0 Fr Gr

Hl JlC 0 0

(1.7)

It turns out that in order to get an approximation of the systemG(s) in theweighted norm (1.6) one needs only to compute the(2, 2) blocks of the twogramians of the partitioned system (1.7), and perform a balanced truncation ofthe subsystemG(s) based on these “sub-gramians"; details are given below.

Closed-loop model reduction problem. In this case, the given plant is part ofa closed-loop system with controllerK(s), i.e., in the simplest case the overall

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system is characterized by the transfer function

φ(G,K) := G(s)(I + K(s)G(s))−1.

A reduced order systemG(s) is sought such that the error

‖φ(G,K) − φ(G,K)‖is kept small or minimized. Alternatively, one may also wish to find a reducedorderK(s) such that‖φ(G,K)− φ(G, K)‖, or even‖φ(G,K)− φ(G, K)‖,is kept small or minimized. Let

G =[

A BC 0

], K =

[F GH 0

],

be the system quadruples, then a realization(At,Bt,Ct) for the compoundφ(G,K) is given by

[At Bt

Ct 0

]=

A −BH

GC FB0

C 0 0

. (1.8)

In order to get an approximation of the systemG(s) in the sense that theclosed-loop norm is minimized, one can again use the(1, 1) blocks of the twogramians of the partitioned system (1.8), to obtain a balanced truncation of thesubsystemG(s).

Gramians for open- and closed-loop systems. Both problems above resultin model reduction of systems with structured state-space matricesAt,Bt andCt. We now explain whyonlyappropriately defined sub-blocks the full grami-ans of this system need to be computed. Given the system(A,B,C), we writethe transfer functionG(s) in the form G(s) = Gyx0(s) [sI−A]Gxu(s),whereGyx0(s) = C(sI − A)−1, andGxu(s) = (sI − A)−1B. Gxu canbe interpreted as the input-to-state transfer function whileGyx0 as the state-to-output transfer function. It is well known that the gramians of this system canbe expressed in the frequency domain as

P(G) =12π

∫ ∞

−∞Gxu(iω)GT

xu(−iω)dω,

Q(G) =12π

∫ ∞

−∞GT

yx0(−iω)Gyx0(iω)dω.

In the case of input weighting,Wr(s), the transfer function of the com-posite system is modified toG(s)Wr(s). The input-to-state transfer functionthus becomesGxu(s)Wr(s), while the state-to-output transfer function re-mains the same. It should be noted thatstatehere refers to the state of the


system of interest, namely,G(s). Consequently, the controllability gramian ismodified toP(Gr), whereGr = GWr, while the observability gramian re-mains the sameQ(GWr) = Q(G). Similarly, for output weighting,Wl(s),the controllability gramian remains unchangedP(WlG) = P(G), while theobservability gramian is modified toQ(Gl), whereGl = WlG. Along thesame lines it follows that for both input and output weightingWr, Wl, wehaveP(WlGWr) = P(GWr) andQ(WlGWr) = Q(WlG).

We now turn our attention to the definition of gramians for closed-loop sys-tems. Consider the systemG(s) in the usual feedback loop together with thecontrollerK(s); the transfer function isφ(G,K) as defined above. This canbe considered as a weighted composite system with input-weightingWr(s) =(I + K(s)G(s))−1. The gramians of the systemG(s) with respect to the givenclosed-loopwith the controllerK(s), are now defined as shown above for theinput weighting case. Similarly, the gramians of the controllerK(s) in thesame closed loop, can be defined by noting that the transfer function of inter-est isK(s)G(s) (I + K(s)G(s))−1. Consequently, this can be considered asa weighted system with input weightingWr(s) = G(s) (I + K(s)G(s))−1.The resulting gramians ofK(s) with respect to the given closed-loop, can beused to reduce the order of the controller in the closed loop.

The question now is whether there are Lyapunov equations that can be usedto solve for the weighted and closed-loop gramians. This is indeed so. Weonly describe here the case of input weighting. Let(At,Bt,Ct) be a statespace realization ofK(s)Wr(s). It can be shown that

P(KWr) =[

In 0]P(At,Bt)

[In

0

],

Q(KWr) =[

In 0]Q(Ct,At)

[In

0

],

whereP(At,Bt) satisfies the Lyapunov equation

AtP(At,Bt) + P(At,Bt)ATt + BtBT

t = 0,

and similarlyQ(Ct,At) satisfies

ATt Q(Ct,At) +Q(Ct,At)At + CT

t Ct = 0.

In conclusion, the model reduction problems we are faced with involvegramians that have special structure coming from (i) the embedding into alarger system with a weighting or with a controller, (ii) the polynomial struc-ture of a mechanical or structural system, and (iii) the additional constraintsimposed on the system such as passivity.

A number of technical challenges must be overcome in order to solve theseproblems. These include: (i) construction of efficient algorithms to exploit

14 DDDAS

the above block structure to compute only the relevant sub-gramians and toformulate numerical schemes that converge to them or approximate them, (ii)deriving error estimates for the balanced truncation of the sub-system and oftheir approximate schemes, and (iii) designing iterative schemes that producea stabilizing controller for a given high-order plant with guaranteed perfor-mance. This should be achieved by successive reduction of the order of theplant and the controller in the closed-loop.

4.2 Second Order Problems

Models of mechanical systems are often of the type (1.0), whereM = MT ,D = DT andK = KT are respectively the mass, damping and stiffness matri-ces, since this represents the equation of motion of the system. In the Laplacedomain, thecharacteristic polynomial matrixP(s) := Ms2 + Ds + K andthe transfer functionG(s) = CP−1(s)B appear and the zeros ofdet(P(s))are also known as thecharacteristic frequencies or polesof the system. Usingan extended stateξ =

[xT xT

]T, this can be linearized to a generalized

state-space model{E ,A,B, C} :[D MM 0

]︸︷︷︸

E

ξ =[ −K 0

0 M

]︸︷︷︸

A

ξ +[

B0

]︸︷︷︸

B

u, y =[

C 0]

︸︷︷︸C

ξ. (1.9)

SinceM is invertible, we can also transform this model to the standard state-

space form{E−1A, E−1B, C}, whereE−1A =[

0 I−M−1K −M−1D

]and

E−1B =[

0M−1B

]. The controllability gramianP and observability gramian

Q of the state-space model{E−1A, E−1B, C} can be computed via the solutionof the Lyapunov equations :

E−1AP+PATE−T +E−1BBTE−T = 0, QE−1A+ATE−TQ+CTC = 0.(1.10)

Moreover, ifC = BT , the transfer function and model (1.9) are symmetricand the gramians of the state-space model are defined in terms of the solutionG of a single generalized Lyapunov equation :

AGE + EGA+ BBT = 0, P = G, Q = EGE . (1.11)

Most model reduction methods use a projection to build the reduced-ordermodel: given a generalized state-space model{E ,A,B, C}, the reduced-ordermodel is given by{ZTEV,ZTAV,ZTB, CV}, whereZ andV are matricesof dimension2N×k, with k the order of the reduced system. The widely usedbalanced truncation technique constructsV andZ := E−1W such that (1.5) issatisfied withΣ2

+ thek × k matrix containing the largest eigenvalues ofPQ.


This technique cannot be applied directly to a second order system since, ingeneral, the resulting reduced order system is no longer a second order system.There is a need foradaptivetechniques to find reduced order models of secondorder type. The idea is to choosek = 2n and to restrict the projection matricesV andW to have a block diagonal form, where each block isN × n :

W =[

W11 00 W22

], V =

[V11 00 V22

], (1.12)

which turns out to be asufficientcondition for obtaining an appropriate reducedorder model. Notice that for the symmetric case we have automaticallyV =W. If we want to impose a block diagonal form, we can, for example, relaxthe equality conditions in (1.5).

In this context, the following technical challenges must be addressed: (i) usea relaxation that can be dealt with using Tracemin like optimization methods,such as:

min trace VT11[P−1]11V11 + VT

22[P−1]22V22; (1.13)

s.t. VT11[Q]11V11 = In, VT

22[Q]22V22 = In

(ii) define alternative gramians of dimensionN ×N for second order systemsand analyze their use for model reduction, and (iii) analyze modal approxima-tion based on the polynomial matrixP(s). One then computes the right andleft eigenvectorsP(λi)xi = 0 andyT

i P(λi) = 0 via the linearized eigenvalueproblem : [

λiD + K λiMλiM −M

] [xi

λixi

]= 0,

[yT

i λiyTi

] [λiD + K λiM

λiM −M

]= 0,

and selectsn of them to form the projection matricesV11 andW11.

4.3 Passivity Preserving Model Reduction

A system is passive if it does not generate energy, and in circuit simulationsuch systems are of great importance. Model reduction schemes that preservepassivity are essential in this application. Existing methods can be divided intwo categories –positive real balancingdue to Ober (Ober, 1991), and struc-tural methods such as the work of Pileggi and Rohrer on the one side (Odaba-sioglu et al., 1998) and Feldmann, Freund, and Bai on the other (Feldman andFreund, 1995; Freund, 1999; Bai et al., 1997; Bai et al., 1998; Bai and Freund,2000). See also the work of White and co-workers (Li et al., 1999; Kamonet al., 2000).

16 DDDAS

Positive Real:It turns out that a real system is passive if its transfer functionG(s) = C(sIn−A)−1B + D is positive real, which means thatG is square(m = p) and analytic in the open right halfplane with withReG(z) ≥ 0 forRe(z) > 0. Thespectral zerosof G are the members of the setSG := {z ∈C : det[G(z) + GT (−z)] = 0}.

Here, we assume that the system is passive and thatD := D+D∗ is positivedefinite, soD = W∗

oWo with Wo nonsingular. It is desirable to place condi-tions on the transfer functionG(s) := C(sI−A)−1B of the reduced systemΣthat provide for the inheritance of passivity from the original system. The ap-proach taken here is motivated by recent results of Antoulas (Antoulas, 2002)that give conditions on sets of interpolation points that assure passivity of thereduced model. Let

A :=

A B

−AT −CT

C BT D + DT

and E :=

I

I0

(1.14)

Then the spectral zeros ofG are the set of all (finite) complex numbersλ suchthatRank(A− λE) < 2n + p, i.e. the finite generalized eigenvaluesσ(A, E).It is easily seen thatλ ∈ SG ⇒ −λ ∈ SG.

We aim to compute a selected invariant subspace of(A, E), and from this,to construct a reduced model that satisfies the appropriate interpolation condi-tions. SupposeAX = EXR is a partial real Schur decomposition for the pair(A, E). ThusX TX = I andR is real and quasi-upper triangular.

The following result from (Sorensen, 2002) is the foundation of the con-struction. LetX T := [XT ,YT ,ZT ] be partitioned in accordance with theblock structure ofA. It can be shown thatXTY is Hermitian and negativesemidefinite with bothX andY full rank. Moreover,

XTATY + YTAX = ZTDZ (1.15)

CX + BT Y = −DZ. (1.16)

Equations (1.15) and (1.16) are closely related to the Positive Real Lemma(PRL) and projections of these result in sufficiency conditions for PRL beingsatisfied by the reduced model. The matricesV andW will be constructedfromX andY. The invariant subspace is defined by specifying the eigenvaluesof R to be either all in the open right half-plane or all in the open left half-plane.If we constructV,W such thatX = VX andY = WY with WT V = Ithen

A B−AT −CT

C BT D + DT

X

YZ

=

X

Y0

R, (1.17)


whereA := WT AV, B := WTB, C := CV. This shows that the spectral

zerosSG are a subset of the spectral zerosSG of the original system. Passivityof the reduced follows from equation (1.15) and ( 1.16) after multiplication onthe left and right byX−1 andX−T (a certain construction derived from theSVD of XTY givesY = −X).

It is shown in (Sorensen, 1992) that the following two Riccati equations

ATQ+QA = −(CT −QB)D−1(C−BTQ), (1.18)

AP + PAT = −(B− PCT )D−1(BT −CP)

are satisfied withQ = YX−1 andP = XY−1 when alln spectral zeros in theleft (or right) open halfplane are interpolated. Reduced order Riccati equationsare satisfied by the reduced model. The minimal solution to either equation isobtained if all interpolated spectral zeros are either all in the left or all in theright half-plane, and balancing transformations can be constructed from these.

In this context, one needs to derive conditions for selecting the subset ofspectral zeros to be interpolated in order to give the best model of specifiedorderk possible from this construction. In (Sorensen, 2002) it is shown thatapplying an implicitly restarted Arnoldi (IRA) method on a Cayley transformof (A, E) to compute an invariant subspace gave an efficient reduction. Thisrequires two sparse direct matrix factorizations ofA + µI andA − µI. Itwould be useful to develop criteria (either rigorous or heuristic) to selectµto best accelerate the convergence of the IRA iteration. It would also be use-ful to develop an approximate shift-invert preconditioner and other specializedschemes specific to this problem. The reduction scheme described above canbe carried out regardless of the positive real condition, but there may be spec-tral zeros on the imaginary axis. When the interpolated spectral zeros are allin the open left half-plane, thenR is stable andAX + BZ = XR indicatesthatF := ZX−1 is stabilizing sinceA + BF = XRX−1 whenX is of ordern. WhenX is n × k, the restriction ofA + BF to Range(V) is stabilized ifF = WT , since(A + BZWT )V = VR, whereR is similar toR. There isa need for appropriate selection criteria on the spectral zeros that will assurethatF := ZWT is actually completely stabilizing. Finally, there is a need foran approximate Riccati solver for equations of the type (1.18) that will giveapproximate solutionsPk andQk in low rank factored form.

5. Time Variance and Adaptivity

Time variance and adaptivity can be either due to actual time variation ofthe parameters of the system, or due to an external iteration used to adaptivelytune the reduced order system. We first explain how to cope with generaltime-varying systems and show how these methods can also be used for time

18 DDDAS

invariant or for periodic systems. We then discuss systems that are iterativelyadapted within an outer tuning loop.

5.1 Time Varying Systems

Linear discrete time-varying systems are described by systems of equations :

xk+1 = Akxk + Bkuk, yk = Ckxk (1.19)

with input uk ∈ Rm, statexk ∈ R

N and outputyk ∈ Rp that are square-

summable. Using the recurrence (1.19) over several time steps, and restrictingthe inputs to be non-zero only in the interval[ki, k) (i.e., the “past"), one thenobtains an expression for the outputs in the interval[k, kf ] :26664

yk

yk+1

...ykf

37775 = OkCk

26664

uk−1

uk−2

...uki

37775 , Ok =

26664

Ck

Ck+1Ak

...Ckf

Φ(kf , k)

37775, CT

k =

26664

BTk−1

BTk−2A

Tk−1

...BT

kiΦ(k, ki−1)T

37775 .

The finite window Hankel map relating inputs in[ki, k) to outputs in[k, kf ]is the productH(k, ki, kf ) = OkCk of the observability matrixOk over[ki, k)and the controllability matrixCk over [k, kf ]. These two matrices are time-varying extensions ofO andC and obviously satisfy the recurrences :

Oj =[

Cj

Oj+1Aj

], Cj+1 =

[Bj AjCj

],

which construct the controllability matrix forward fromki to k and the ob-servability matrix backward fromkf to k. Let us compute these recurrencesusing low rank approximations at each time step, according to the followingrecursive scheme : we construct matricesSc(k),So(k)∈R

N×n satisfying

CjCTj = Sc(j)ST

c (j) + Ec(j)ETc (j), OT

j Oj = So(j)STo (j) + Eo(j)ET

o (j),

whereEc(j) andEo(j) are small in norm, and wherePj := CjCTj , Qj :=

OTj Oj are in fact the time varying gramians of the system over the correspond-

ing time windows. The matricesSc(k) andSo(k) are obtained recursively froma singular value decomposition of the type[

Cj

STo (j + 1)Aj

] [B` A`Sc(`)

]= U(i)Σ(i)V(i)T . (1.20)

PartitioningU(i) =[U1(i) U2(i)

], V(i) =

[V1(i) V2(i)

]one then

updates the factors as follows

Sc(` + 1) =[

B` A`Sc(`)]V1(i), (1.21)

STo (j) = UT

1 (i)[

Cj

STo (j + 1)Aj

],


where the indicesj, ` depend oni (Chahlaoui and Dooren, 2002). For continuous-time systems

x(t) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t), (1.22)

one can still apply the same ideas by converting (1.22) to a discrete-time system(1.19) via, for example, the Euler method. One can then view (1.21) as anintegration formula for computing the gramiansP andQ (Gugercin et al.,2003).

In this context, we need to derive bounds for the low rank approximations ofthe gramians and Hankel map of the low order model{E−1

k Ak, E−1k Bk, Ck},

where{Ek, Ak, Bk, Ck} .= {STo (k)Sc(k), ST

o (k)AkSc(k), STo (k)Bk, CkSc(k)}.

This method can be applied to periodic systems and time-invariant systems, inwhich case we need to analyze the asymptotic stability of the obtained re-duced order model. We need to analyze the connection, and relative merit, ofthe above updating method and the Smith method, apply this to second ordertime-varying systems and use possibly other integration methods, and analyzethe convergence of the integration scheme for time-invariant systems and its“tracking” capabilities for (slowly) time varying systems.

5.2 Iterative Schemes

Reduced order stabilization of a large plantG(s) amounts to construct-ing a small order controllerK(s) such that the closed loop transfer functionφ(G, K) = G(s)(I + K(s)G(s))−1 is stable and such that at the same timesome performance criterion is optimized. We discuss the issues involved andthe ensuing problems in terms of an example.

Example: Iterative control of a CD player model. An important part of a CDplayer is a control system which makes it less sensitive to external shocks. Themechanism involved is a swing arm on which a lens is mounted. As the discrotates the reader tries to follow the spiral track focusing at the same time. Thegoal is to find a low-cost controller that makes the system faster and less sensi-tive to external shocks. Although robustness issues are important, to keep thepresentation simple we focus here on the control of a single model of the CDarm. To achieve the above goal, a detailed model of the vibrational behaviorof this electro-mechanical system is needed; for this purpose a finite elementmodel can be built with few hundred vibrational modes, thus resulting in amodel whose order is twice the number of modes. The system has two inputs,namely, actuation of the arm and of the lens, and two outputs, namely, trackingerror and focus error.

Most often a crude low-order (PI) controller is available initially. In orderto come up with a simple controller that minimizes an appropriate norm of thetransfer function between the inputs and the tracking/focusing errors respec-

20 DDDAS

tively, an iterative procedureis used. Let us denote the original high ordermodel byG0(s), the original PI controller byK0(s), and the resulting perfor-mance criterion byJ (G0,K0) (which may be theH2 or theH∞ norm of theerror transfer function). Using aclosed-loopmodel reduction procedure,G0

is reduced toG1 so that the relative error

γ1 = 1− J (G1,K0)J (G0,K0)

� 1

is small. The second part of the first step consists of optimal controller syn-thesis for the nominal plantG1, which yieldsK∗

1; againH2 orH∞ methodscan be used. The third part of the first step consists in usingclosed-loopmodelreduction ofK∗

1(s) to K1 so thatδ1 = J (G1,K1) is close to the optimalperformanceδ∗1 = J (G1,K∗

1). We are now ready to proceed with the secondstep which repeats the procedure based onG1, K1.

The point of the iteration is to successively reduce the order of the highorder system until the corresponding optimal controller has low enough orderas well as the specified performance properties. Thus, at each reduction step,performance is checked by means of the indicesγi, δ∗i andδi.

The rationale of this approach is that the controller should be designed fora small order plantGi(s) using dense matrix techniques. However, this con-troller will only have a good performance for the original plantG(s) if theapproximationGi(s) is obtained using the closed loop error‖φ(G, Ki) −φ(Gi, Ki)‖. These two criteria depend of course on each other, which is whyan outer iteration is needed.

The ingredients needed to make such an outer iteration tractable are pre-cisely addressed in previous sections. The models of the closed loop systemsφ(G, Ki) are as given by formula (1.7), where{Fi,Gi,Hi} are the systemtriples of thei-th iterateKi(s) of the controller. This is a structured problemwith coefficient matrices that evolve with each iteration step. The goal is to ex-ploit these features, especially when the time variation becomes smaller (i.e.,upon convergence). A second ingredient is the construction of an original sta-bilizing controller. Our previous work on the construction of stabilizing con-trollers using invariant subspace ideas and Riccati iterations for large sparsesystem models (Rao et al., 2000) is relevant here. A third ingredient is theconvergence test, in which one should assess whether or not the closed loopsystem is stable or if other structural properties are satisfied or not.

In this context, one needs to analyze the convergence of the closed loop iter-ation and find global/local convergence strategies These ideas can be applied tofinding reduced order models of plants where certain properties are imposed,such as stability, passivity, or second order model constraints. Finally, thisproblem can be embedded in a model reduction problem with time-varying


coefficients. Therefore it is possible to take advantage of this fact in order tocompute the new plant/controller byupdatingthe previous ones.

6. Simulation and Validation Support

An important aspect of model reduction and control is to develop a compre-hensive environment for validation and refinement. Instrumenting full-scalesystems with control mechanisms can be an expensive proposition. For thisreason, control techniques must be extensively tested in a simulation environ-ment prior to scale model testing. We have developed an extensive simulationenvironment for complex structures and demonstrated the power of this envi-ronment in the context of several structural impact response simulations (thetragic Pentagon crash on 9/11 is illustrated in Figure 1.1).

With a view to understanding the nature, extent, and cause of structural dam-age, we have used our simulation environment to perform high-fidelity crashsimulations on powerful parallel computers. A single simulation instance il-lustrated in Figure 1.1 with one million nodes (3 d.o.f. per finite element node)over 0.25 seconds of real time takes over 68 hours on a dedicated 8 processorIBM Regatta SMP. A coarser model with 300K nodes over 0.2s of real timetakes approximately 20 hours of simulation time. While we are also workingon improving the underlying solvers in our simulation, these computationalrequirements provide strong motivation for effective error-bounded model re-duction. In this context, development of effective preconditioners and itera-tive solvers, meshing and mesh adaptation techniques, parallel algorithms, andsoftware development and integration pose overarching technical challenges.

7. Concluding Remarks

Model reduction is an extremely important aspect of dynamic data-drivenapplication systems, particularly in resource-constrained simulation and con-trol environments. While considerable advances have been made in model re-duction, simulation, and control, significant challenges remain. In conjunctionwith the research activity in areas of sensor networks, actuation mechanisms,and ad-hoc wireless networks, techniques for model reduction and control holdthe key to fundamental advances in future generation systems.

22 DDDAS

(a) Aerial view of damage to thePentagon building.

(b) Simulated aircraft impact onRC columns in the building.

Figure 1.1. High fidelity simulation of the Pentagon crash performed in our group has yieldedsignificant insights into structural properties of reinforced concrete and design of resilient build-ings.

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