1

Optimal Sensor and Actuator Selection using Balanced Model ReductionKrithika Manohar, J. Nathan Kutz, Member, IEEE, and Steven L. Brunton, Member, IEEE

Abstract—Optimal sensor and actuator selection is a centralchallenge in high-dimensional estimation and control. Nearly allsubsequent control decisions are affected by these sensor/actuatorlocations, and optimal placement amounts to an intractablebrute-force search among the combinatorial possibilities. In thiswork, we exploit balanced model reduction and greedy optimiza-tion to efficiently determine sensor and actuator selections thatoptimize observability and controllability. In particular, we deter-mine locations that optimize scalar measures of observability andcontrollability via greedy matrix QR pivoting on the dominantmodes of the direct and adjoint balancing transformations.Pivoting runtime scales linearly with the state dimension, makingthis method tractable for high-dimensional systems. The resultsare demonstrated on the linearized Ginzburg-Landau system, forwhich our algorithm approximates known optimal placementscomputed using costly gradient descent methods.

Index Terms—optimal control, balanced truncation, sensorselection, actuator selection, observability, controllability.

I. INTRODUCTION

Optimizing the selection of sensors and actuators is oneof the foremost challenges in feedback control [1]. For high-dimensional systems it is impractical to monitor or actuate ev-ery state, hence a few sensors and actuators must be carefullypositioned for effective estimation and control. Determiningoptimal selections with respect to a desired objective is anNP-hard selection problem, and in general can only be solvedby enumerating all possible configurations. This combinatorialgrowth in complexity is intractable; therefore, the placementof sensors and actuators are typically chosen according toheuristics and intuition. In this paper, we propose a greedyalgorithm for sensor and actuator selection based on jointlymaximizing observability and controllability in linear time-invariant systems. Our approach (see Fig. 1) exploits low-ranktransformations that balance the observability and controlla-bility gramians to bypass the combinatorial search, enablingfavorable scaling for high-dimensional systems.

To understand the challenges of sensor and actuator place-ment for estimation and control, we will first consider optimalsensor placement, which has mostly been used to reconstructstatic signals. The primary challenge of sensor selection is thatgiven n possible locations and a budget of r sensors, thereare combinatorially many,

(nr

), configurations to evaluate in a

brute-force search. Fortunately, there are heuristics that employgreedy selection of sensors based on maximizing mutualinformation [2] and information theoretic criteria [3]. Anotherpopular approach relaxes sensor selection to a weighted convexcombination of possible sensors [4], [5], [6], typically solved

The authors thank Bing Brunton, Eurika Kaiser, and Josh Proctor forvaluable discussions. SLB acknowledges support from AFOSR Grant FA9550-18-1-200. JNK acknowledges support from AFOSR Grant FA9550-19-1-0011.KM acknowledges support from NSF MSPRF Award No. 1803289.

K. Manohar was with the Department of Applied Mathematics, Universityof Washington, Seattle, WA, 98195 USA e-mail: kmanohar@uw.edu.

J. N. Kutz and S. L. Brunton are with University of Washington.

2015

105

0-5

-10-15

-20

Optimal Feedback Control(sparse sensors/actuators)

Balanced Sensor/Actuator

Selection

Complex System

Offline learning

SensorsActuators

Time

Space

Fig. 1: Schematic of balanced sensor and actuator selectionfor the optimal control of a high-dimensional system.

using semidefinite programming. Both heuristic approachesoptimize submodular objective functions [7], which boundthe distance between heuristic and optimal placement. Someobjectives, such as those based on the quality of a Kalmanfilter, are not submodular [8]. Alternatively, sparsity-promotingoptimization can be used to determine sensors and actua-tors [9], [10], [11], although non-differentiability of sparsitypromoting terms motivates other optimization techniques [12].

Even such heuristics cannot accommodate the large dimen-sion of many physical models, such as in fluid dynamics.Fortunately, high-dimensional systems often evolve accordingto relatively few intrinsic degrees of freedom. Thus, it is possi-ble to leverage dimensionality reduction to strategically selectsensors. One approach to place point sensors [13] computesthe empirical interpolation points via EIM [14] correspondingto the proper orthogonal decomposition (POD) [15] of data,to determine important locations in state space.

For systems with actuation, it is necessary to simultaneouslyconsider the placement of sensors and actuators, since themost observable and most controllable subspaces are oftendifferent. Sensors and actuators for optimal feedback controlare generally placed along the most observable and control-lable directions, respectively [16], [17], [18], [19], [7], usingobjective functions based on the associated observability orcontrollability gramians. Standard metrics for evaluating acertain sensor/actuator configuration include the H2 norm [20],[16], a measure of the average impulse response, and theH∞ norm to measure the worst case performance. A chiefdrawback is the need to recompute the controller with eachnew configuration of sensors and actuators given by eitherthe gradient minimization computation or brute-force searches.Moreover, these methods do not exploit the state-of-the-art inmodel reduction to optimize sensor and actuator placement.

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Contribution. This work develops a scalable sensor and actu-ator selection algorithm based on balanced truncation [21], inwhich modes are hierarchically ordered by their observabilityand controllability. We use empirical interpolation of the low-rank balanced representation to find maximally observable andcontrollable states. The resulting locations correspond to near-optimal point sensor and actuator configurations. The qualityof our optimized configurations are evaluated using the H2

norm of the resulting system, which is an average measure ofits output energy. The closed loop H2 norm is more relevantthan open loop metrics for control performance, given a spe-cific H2 cost function. Our approach, when used to optimizethe open loop H2 norm, is agnostic to the specific choice ofcontroller weight matrices, and instead maximizes the input–output energy of the reduced order model. We also show thatit is possible to apply our framework to closed loop systems,demonstrating near optimal sensor and actuator selection incomparison with more expensive iterative closed loop H2

optimization. The runtime scales linearly with the number ofstate variables, after a one-time offline computation of thebalancing transformation, which is less expensive than iterativealternatives. The resulting sensor and actuator configurationsreproduce known optimal locations at a fraction of the costassociated with competing gradient descent methods.

II. PROBLEM SETUP

Consider the following linear time-invariant system with agiven state-space realization

x = Ax + Bu x ∈ Rn,u ∈ Rq (1a)y = Cx, y ∈ Rp, (1b)

with large state dimension, i.e., n � 1. It is assumed thatthe system is stable, and B and C are linear actuation andmeasurement operators that make the system observable andcontrollable. Our objective is to choose a minimal subset ofthese sensors and actuators to obtain a system that is mostjointly controllable and observable. For illustration we beginwith B = C = I which correspond to pointwise sensing andactuation, but in general the subset selection can be adaptedfor arbitrary B and C. This subset selection corresponds tomultiplying inputs and outputs by the selection matrices

SC =[eγ1 eγ2 . . . eγr

]T(2a)

SB =[eβ1

eβ2. . . eβr

]. (2b)

Here ej are the canonical basis vectors for Rn with a unitentry at the selected index j and zeros elsewhere, whereγ = {γ1, . . . , γr} ⊂ {1, . . . , p} denotes the index set of sensorlocations with card(γ) = r. Similarly, actuator selectionindices are given by β = {β1, . . . , βr}. The new measurementand actuation operators are C = SCC and B = BSBrespectively. In the special case B = C = I, the new operatorsC = SCI and B? = ISB select subsets of state inputs andoutputs, and the output would consist of r components of x

y = Cx = [xγ1 xγ2 . . . xγr ]T . (3)

Problem statement: What are the best r-subsets of a givenset of p sensors and q actuators, where r � n?

To answer this question, we first quantify the degree ofobservability and controllability for a given set of sensors andactuators, i.e. for a given choice of C and B. Optimizingover these directly involves a combinatorial search, and thus aheuristic approach is necessary for high-dimensional systems.

A. Observability and controllability

The degrees of observability and controllability for the state-space system (1) are quantified by the observability gramianWo and controllability gramian Wc

Wo =

∫ ∞0

eA∗tC∗CeAtdt, Wc =

∫ ∞0

eAtBB∗eA∗tdt,

(4)which may be visualized as controllable and observable ellip-soids (Fig. 2). These depend on the actuation and measurementoperators, which consist of all states reachable from a boundedinitial state

Ec = {W1/2c x | ‖x‖2 ≤ 1}, (5)

and all states that may be observed

Eo = {W1/2o x | ‖x‖2 ≤ 1}. (6)

Because the gramians depend on B and C, they are often usedto evaluate the observability/controllability of a given sensorand actuator placement. One important evaluation metric isthe H2 norm of a system. It measures the average output gainover all frequencies of the input, or the output energy. For thestate-space system (1) with transfer function G(s) = C(sI −A)−1B, it is given by

‖G‖22 =1

4π2

∫ ∞0

tr(G(jω)∗G(jω))dω. (7)

By the Plancherel theorem, it is also defined in the timedomain by the impulse response yij(t) = Cie

AtBj - theoutput in component i given an impulse in input j,

‖G‖22 =

∫ ∞0

tr(CeAtBB∗eA∗tC∗)dt = tr(CWcC

∗) (8a)

=

∫ ∞0

tr(B∗eA∗tC∗CeAtB)dt = tr(B∗WoB) (8b)

which explicitly relate each gramian to both B and C. Arelated alternative to the average output energy metric is givenby the volumetric measure, the log determinant, denoted

log |CWcC∗|, log |B∗WoB|, (9)

which are the logarithms of the geometric mean of the axesof the ellipsoid skewed by B or C, by comparison the trace isthe arithmetic mean. This metric is introduced by Summers etal [7] to place actuators using a greedy optimization schemefor the submodular objective function

B? = argmaxB

log |CWcC∗|. (10)

For H2 optimal control it is desirable to minimize the averagegain from stochastic disturbance w to control output z(s) =

3

G(s)w(s), namely, minimizing ‖G‖2. Several strategies seekto build the controller and choose actuators simultaneously,using expensive gradient optimization schemes. The drawbackof such closed loop metrics is having to recompute thegramians - an O(n3) operation - for every iteration that selectsthe next best actuator. This cubic scaling may be intractablefor high-dimensional systems with large n.

There are cases where optimizing sensors and actuators us-ing the closed loop H2 norm is more relevant for control [20],[16]. By contrast, our approach reverses the strategy byinstead starting from a maximally actuated and sensed optimalcontroller, then seeks a subset of these sensors/actuators topreserve (maximize) the geometric control measure, namely

SC? = argmaxSC

log |SCCWcC∗S∗C |, (11a)

SB? = argmaxSB

log |STBB∗WoBSB |. (11b)

Now, the gramians no longer depend on the optimizationvariable and need only be computed once, and both objectivesare still fundamentally linked to the H2 norm of the system.Critically, we will extract the dominant controllable and ob-servable subspaces from a balanced coordinate transformationof the gramians.

III. BALANCED MODEL REDUCTION

Many systems of interest are exceedingly high dimensional,making them difficult to characterize and limiting controllerrobustness due to significant computational time-delays. How-ever, even if the ambient dimension is large, there maystill be a few dominant coherent structures that characterizethe system. Thus, significant effort has gone into obtainingefficient reduced-order models that capture the most relevantmechanisms for use in real-time feedback control [1].

The goal of balanced model reduction is to find a trans-formation T from state-space (leaving inputs and outputs

unchanged),[

A BC 0

]to[

TAT−1 TBCT−1 0

], such that the

transformed coordinates a = T−1x are hierarchically orderedby their joint observability and controllability. This permits anr-dimensional representation made possible by truncating then− r least observable and controllable states.

The seminal work of Moore in 1981 [21] showed it ispossible to compute this coordinate system Ψ where the con-trollability and observability gramians are equal and diagonal,denoted by the balanced model

a = Φ∗AΨa + Φ∗Bu a ∈ Rn,u ∈ Rq

y = CΨa. y ∈ Rp (12)

Here T−1 , Ψ are direct modes and T , Φ∗, the adjointmodes. The balanced state a is then truncated, keeping onlythe first r � n most jointly controllable and observable statesin ar, so that x ≈ Ψrar. This results in the balanced trunca-

tion model [21] Gr =

[Φ∗rAΨr Φ∗rB

CΨr 0

]. Since gramians

depend on the particular choice of coordinate system, they will

ˆ ˆ

Fig. 2: (top) Illustration of the balancing transformation forgramians. The reachable set Ec with unit control input is shownin blue. The corresponding observable set is shown in red.Under the balancing transformation Ψ, the gramians are equal,shown in purple. (bottom) Sensor and actuator selection basedon balancing transformation.

transform under a change of coordinates. The controllabilityand observability gramians for the balanced truncated systemare

Wc = Φ∗WcΦ, Wo = Ψ∗WoΨ. (13)

The coordinate transformation Ψ that makes the controllabilityand observability gramians equal and diagonal,

Wc = Wo = Σ, (14)

is given by the matrix of eigenvectors of the product of thegramians WcWo in the original coordinates:

WcWo = Φ∗WcWoΨ = Σ2 =⇒ WcWoΨ = ΨΣ2. (15)

The resulting balanced system is quantifiably close to theoriginal system in the H∞ norm in terms of the Hankelsingular values or diagonal entries of Σ

‖G−Gr‖∞ ≤ 2

n∑i=r+1

σi. (16)

In practice, computing the gramians Wc and Wo and theeigendecomposition of the product WcWo in (15) may beprohibitively expensive for high-dimensional systems. Instead,the balancing transformation may be approximated with datafrom impulse responses of the direct and adjoint systems,utilizing the singular value decomposition for efficient ex-traction of the relevant subspaces. The method of empiricalgramians is quite efficient and is widely used [21], [22], [23],[24]. Moore’s approach computes the entire n × n balancingtransformation, which is not suitable for exceedingly high-dimensional systems. In 2002, Willcox and Peraire [23] gen-eralized the method to high-dimensional systems, introducinga variant based on the rank-r decompositions of Wc and Wo

obtained from snapshots of direct and adjoint simulations. It isthen possible to compute the eigendecomposition of WcWo

4

using efficient eigenvalue solvers. This approach requires asmany adjoint impulse-response simulations as the number ofoutput equations, which may be prohibitively large for full-state measurements. In 2005, Rowley [24] addressed this issueby introducing output projection, which limits the number ofadjoint simulations to the number of relevant POD modes inthe data. It is particularly advantageous to use these data-driven methods or low-rank alternating direction methods [25]to approximate the gramians when there are fewer than fullmeasurements and actuation of the state.

IV. SENSOR & ACTUATOR OPTIMIZATION VIA QR PIVOTING

We now describe an efficient matrix pivoting algorithmto optimize the log determinant over the choices of sensorsand actuators. The representation of the gramians in balancedtruncation coordinates plays a crucial role.

A. Matrix volume objective

Recall the goal of optimizing a set of r sensors and actuatorsout of a fixed set p and q possible choices. The budget rdetermines the balancing rank truncation, which necessarilymust be less than both p and q. Our sensor-actuator selectioncan be regarding as interpolating this rank-r representation,that is, choosing locations or interpolation points that areheavily weighted in the dominant r balanced modes.

Summers et al [7] show that it suffices to only considercontrollable or observable subspaces for selecting sensors andactuators using the log determinant objective. Thus, we cansubstitute rank-r balanced approximation of the gramians, Wc

and Wo, into the log determinant objective

C? ≈ argmaxSC

log |SCCΨrΣrΨ∗rC

TSTC |

= argmaxSC

|SCCΨr|2 · |Σr|

= argmaxSC

|SCCΨr|. (17)

This result follows from the monotonicity of logarithmsand the product property of determinants, then omitting theterm that is independent of the sensors, detΣr. Likewise,in the actuator case, the objective argmaxSB log |BTWoB|simplifies

B? = argmaxSB

|Φ∗BSB |. (18)

Consider for now the case of sensor placement. The absolutedeterminant is a measure of matrix volume, and SC is arow selection matrix. The transformed objectives may beviewed as a submatrix volume maximization problem, whichinvolves choosing the optimal r-row selection of CΨr with thelargest possible determinant. Finding this optimum is an NP-hard, intractable combinatorial search over all possible r-rowsubmatrices of CΨr. However, it can be optimized greedilyand efficiently via one-time matrix QR factorization requiringO(pr2) and O(qr2) operations, as described next.

B. QR pivoting algorithm

The QR factorization with column pivoting is a greedysubmatrix volume optimization scheme that we will use toconstruct C and B, given Ψr and Φr. The pivoted QR factorsany input matrix V ∈ Rr×p into a unitary matrix Q, andupper-triangular matrix R, and column permutation matrix Pso that the permuted matrix VP is better conditioned than V

VP = QR. (19)

However, we seek a well-conditioned row permutation ofCΨr. Consider the input V = (CΨr)

∗ to the QR factoriza-tion, and the leading r×r square submatrices of the permutedinput on both sides of (24),VP and T[

VP | ∗]= [Q]

[T | ∗

]. (20)

Each iteration of pivoting works by applying orthogonal pro-jections to successive columns of V to introduce subdiagonalzeros in R. For our purposes, P plays the crucial role: ateach step P stores the column “pivot” index of the columnselected at each iteration to guarantee the following diagonallydominant structure in R

|Rii|2 ≥k∑j=i

|Rjk|2; 1 ≤ i ≤ k ≤ p. (21)

Observe that the quantity of interest, the determinant ofthe row-selected submatrix VP corresponding to the subsetselection of measurements, now satisfies

|VP | = |Q||T| =r∏i=1

|Tii|, (22)

since Q is unitary and T is upper-triangular. Because thedeterminant is the product of these diagonal entries, it canbe seen that diagonal dominance guaranteed by the pivotingimplicitly optimizes the desired submatrix determinant. ThusSC is constructed from the first r columns of P transposed

SC , (P.,j)T , where j : 1→ r. (23)

Actuator selection proceeds similarly to construct a submatrixof r columns of B∗Φr with maximal determinant, using oneadditional QR factorization

(Φ∗rB)P = QR. (24)

The solution SB is precisely the leading r columns of P,SB , P.,j , and we denote by

C = SCC, B = BSB (25)

the new measurement and actuation operators obtained in thismanner.

The QR pivoting routine is a standard tool in scientificcomputing for matrix decomposition and linear least-squaresproblems. We use a block accelerated implementation ofclassical Businger-Golub pivoting [26] in MATLAB. RecentlyQR pivoting was used for interpolating nonlinear terms inEIMs [14], which would otherwise require the evaluation ofhigh-dimensional inner products. In this setting, the interpo-lation point selection operator is analogous to our selection

5

operator SC used with pointwise measurements (C = I).The algorithm can be analyzed in terms of the error betweenthe full state and the interpolant approximation at QR pivotinterpolation points. The interpolation points can now bewritten

y = Cx ≈ CΨrar, (26)

where Ψr are the POD modes of the reduced model, andar are the modal coefficients. Recovering the state using theinterpolant in the POD basis is accomplished with standardleast-squares approximation

x = Ψr(CΨr)−1y = Ψr(CΨr)

−1Cx. (27)

This can be expressed as a projection PC , Ψr(CΨr)−1C

of the true state x into the observable subspace. As we shallsee, the upper bound on the approximation error

‖x−Ψr(CΨr)−1Cx‖2 (28)

is given by ‖(CΨr)−1‖2 = 1/|Trr|. The connection between

the latter and maximizing the submatrix determinant can bemade explicit in terms of the Hankel singular values of G.

V. ANALYSIS

The best approximation to the state in the span of the directmodes is given by x? , ΨrΦ

∗rx in the ideal measurement

scenario y = x, i.e. C = I. Here the approximation is boundedby the well-known balanced truncation error

‖x− x?‖2 ≤ 2(σr+1 + · · ·+ σn), (29)

where σk are the Hankel singular values, the diagonal entriesof the balanced gramian Σ (14). The analysis of empirical QRinterpolation in the balanced modes begins with an establishedresult for measurements selected using QR, which states that‖(CΨr)

−1‖2 at most grows as√pO(2r).

Lemma 1 (Drmac & Gugercin [14]): The spectral norm of(SU)−1 where S is computed from the QR factorization (23)of the full-rank matrix U ∈ Rp×r is bounded above

‖(SU)−1‖2 ≤√p− r + 1

σmin(U)

√4r + 6r − 1

3. (30)

We generalize this result to the setting of arbitrary linear mea-surements and actuation, by analyzing the residual between thestate and its interpolation in balanced coordinates. Note thatthe residual between the state and its projection into balancedmodes v = x− x? satisfies

PCv = PCx−Ψr(CΨr)−1CΨrΦ

∗rx? = PCx− x?.

The interpolation error from QR pivot selection satisfies

‖x− PCx‖2 = ‖(v + x?)− (PCv + x?)‖2 = ‖(I− PC)v‖2≤ ‖PC‖2‖x− x?‖2≤ ‖Ψr‖2‖(CΨr)

−1‖2‖C‖2‖x− x?‖2.

Substituting (29),(30) above yields the following result.

Theorem 2: The approximation error from interpolating QR-selected observations (23) in balanced truncated modes is

controlled by the discarded Hankel singular values and thenorms of the given measurements and direct modes

‖x− PCx‖2 ≤‖C‖2‖Ψr‖2σmin(CΨr)

√pO(2r)

n∑i=r+1

σi. (31)

The term ‖C‖2 = ‖C‖2 results from information loss whenC 6= I. An analogous result is obtained for actuator selectionby considering the dual problem of estimating the adjoint statefrom actuation matrix B - which is now the measurementoperator of the adjoint system. The resulting projection op-erator, PB , Φr(B

∗Φr)−1B∗, now projects on the span of

the adjoint modes Φr. Making appropriate substitutions of PBin the above results yields the following.

Corollary 1: The approximation error from interpolatingQR-selected observations (25) of the adjoint state in balancedtruncated modes is controlled by the discarded Hankel singularvalues and the norms of the given actuators and adjoint modes

‖z− PBz‖2 ≤‖Φr‖2‖B‖2σmin(Φ∗rB)

√qO(2r)

n∑i=r+1

σi. (32)

We now relate the approximation error bounds using QR pivotsensors and actuators to the log determinant objectives.

Theorem 3: Given direct modes Ψr, QR pivot sensors Cguarantee the following lower bound for the log determinant

r log9σ2

min(CΨr)

(p− r + 1)(4r + 6r − 1)+

r∑i=1

log σi ≤ log |CWcCT |.

Proof: Noting the relationship between the singular values ofa matrix and its QR factorization, we can express |CΨr| interms of the diagonal entries of its R factor

|CΨr| =r∏i=1

σi(CΨr) =

r∏i=1

|Tii| ≥ |Trr|r, (33)

due to nondecreasing σi(CΨr) for increasing i. By squaringthe inequality and multiplying by |Σr| we obtain

T 2rrr · |Σr| ≤ |CΨr|2 · |Σr||CΨrΣrΨ

∗rC

T | = |CWcCT |,

where taking logarithms yields

r log T 2rr +

r∑i=1

log σi ≤ log |CWcCT |.

Because ‖(CΨr)−1‖2 = 1/|Trr|, the upper bound (30) in

Lemma 2 is the inverse lower bound for |Trr|, which can nowbe substituted above to obtain the final result.

An analogous lower bound can be obtained for the objectiveusing QR pivot actuators by appropriately substituting B, Rand adjoint modes Φr in the above proof.

Corollary 2: Given adjoint modes Φr, B satisfies thefollowing lower bound for the log determinant

r log9σ2

min(Φ∗rB)

(q − r + 1)(4r + 6r − 1)+

r∑i=1

log σi ≤ log |BTWoB|.

6

log |CWcC∗|

tr CWcC∗

Fig. 3: QR pivot sensors (red) greedily maximize the logdeterminant objective and H2 norms (trace) over all possibleselections of 7 sensors out of 25 (blue).

VI. RESULTS

We evaluate the selection algorithm in two settings. The firstcompares QR pivot selections with all possible sensor subsetselections in a random state-space model of tractable size. Nextwe consider an application to closed-loop flow control usingLQG control to stabilize unstable Ginzburg-Landau dynamics.The LQG controller with full actuation and sensing is alsotractable, and we approximate the H2 optimal placementscomputed using gradient descent [16] with our QR scheme.

A. Discrete random state space

Our first example investigates sensor and actuator selectionfor random state-space systems with randomized A,B,C.First, we compare the results of QR sensor placement againsta brute-force search across all possible sensor selections fora system with n = 25 states and r = 7 randomizedmeasurements. The log determinant objective (11) is evaluatedfor all possible choices of 7 sensors, since the system issmall enough to explicitly compute the full gramian for all(nr

)= 480, 700 choices of C. These results are binned in

Fig. 3, and compared with the value resulting from our method(red line). The input to the QR scheme, the balancing modes,are computed only once from the full system. The sensorsresulting from our method are observed to be near optimalfor the log determinant, exceeding 99.99% of all others, andalso good substitutes for H2 optimal sensors. On average, ourmethod surpasses 99.8% of possible outcomes with a standarddeviation of 0.85%, over a randomly generated ensemble of500 model realizations. Therefore, QR sensors are closer tooptimal than the analysis suggests.

We now investigate performance on a larger random state-space model with n = 100 states, and likewise initializethe model with randomized actuation and sensing such thatp = q = 100. Figure 4 shows the log determinant objectivethat is being optimized for various sensor and actuator config-urations. The log determinant of the gramian volume is plottedfor the truncated model with QR-optimized sensor and actuatorconfigurations (red circles) and with random configurations(blue violin plots). The truncation level r for the balancedtruncation is chosen to match the sensor and actuator budgeton the x-axis. The QR-optimized configurations dramaticallyoutperform random configurations. As more modes are re-tained, the chosen sensors and actuators better characterize

Fig. 4: Sensor and actuator placement in a random state-space system. The log determinant objective is plotted for QR-optimized sensor-actuator selections (red) and an ensemble of200 random sensor-actuator selections (blue violin plots). Thetruncation level r (also the sensor/actuator budget) varies onthe horizontal axis.

the input–output dynamics, and their performance gap overrandom placement increases over all random ensembles, givingempirical validation of our approach.

Because the system is randomly generated and the dynamicsdo not evolve according to broad, non-localized featuresin state-space, many sensors and actuators are required tocharacterize the system. In particular, this is reflected in theslow decay of Hankel singular values. By contrast, the nextexample is generated by a physical fluid flow model, and hascoherent structure that allow for a more physical interpretationof sensor and actuator placements with enhanced sparsity.

B. Linearized Ginzburg-Landau with stochastic disturbances

We consider the closed-loop linearized Ginzburg-Landaumodel evolving velocity perturbations in a flow, given acontroller with full actuation and sensing, which is often notfeasible in practice. The equations modeling the plant dynam-ics are unstable because the system matrix has eigenvalues inthe right half plane. The dynamical system matrix A is formedfrom Hermite pseudospectral discretization of the linearizedGinzburg-Landau operator

A , −ν ∂∂ξ

+ µ(ξ) + β∂

∂ξ2. (34)

The spatial grid ξ ∈ Rn is discretized at the n = 100 rootsof Hermite polynomials, and ν, β, and µ(ξ) are advection,diffusion and wave amplification parameters. Each ith sensorξs (row of C2) and actuator at ξa (column of B2) are weightedby Gaussian kernels and the trapezoidal integration weights M

C2i ,

[e− (ξ−ξs)2√

2σ

]TM, B2i , e

− (ξ−ξa)2√2σ . (35)

The linear quadratic Gaussian (LQG) controller stabilizesthe dynamics by minimizing the H2 optimal cost functionJ(x,u) = xT Qx+uT Ru, where Q and R are user-specifiedweight matrices. The output u of the LQG controller, givenby [

˙xu

]=

[A−B2F− LC2 L

−F 0

] [xy

], (36)

7

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 1256.3

37.2

31.8

29.3

27.8

(a)QR

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 1250.1

40.1

31.4

29.3

28.1

(b)QR

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 1246.1

34.2

30.8

28.7

27.4

(c)H2 opt

Fig. 5: Sensor (×) and actuator (◦) placement for linearizedGinzburg-Landau. Each row corresponds to the optimizedplacement for budgets of 1-5 sensors and actuators. Placementsbased on QR pivoting of balanced truncated modes (a) closelyapproximate the H2 norms of the placements determined usinggradient descent (c). The QR method can be modified to placesensors and actuators to avoid collocation (b).

stabilizes the dynamics given white noise stochastic distur-bance d and noise n at all sensors and actuators, withcovariances V = 4 · 10−8I and W = I. Since every stateis observed and actuated, each spatial gridpoint correspondsto one ξa and ξs. The idea is to preserve as much of this“ideal” controller as possible using a subset of the originalsensors and actuators

x = Ax + B2SBu + W1/2d (37a)

y = SCC2x + V1/2n, (37b)

which is similar to our original problem formulation. Hencewe can perform balanced model reduction on either impulseresponses or the controller directly, and then QR pivoting tooptimize placements. In this formulation, u is the output andy is the input, which encapsulates the notion of maximizingthe gain from feedback to u to stabilize the dynamics, whichtranslates to the observabillity of (36). Thus we computegramians and adjoint, direct modes of the LQG matricesA , A−B2F− LC2,B , L,C , −F.

We compare our approach to established gradient descenttechniques for computing the H2 optimal controller andsensor-actuator placements simultaneously. The particular al-gorithm for comparison is the optimal placement for thismodel determined using the gradient descent scheme of Chenand Rowley [16].

Their H2 norm optimization scheme permits placement ofsensors and actuators at locations that may not be grid points.

ω = 10−1 ω = 101 ω = 103(a)H2opt

0 0.2 0.4 0.6 0.8 10

0.5

1

-20

-10

0

10

20

(b) QR

0 0.2 0.4 0.6 0.8 10

0.5

1

-20

-10

0

10

20

Fig. 6: LQG gain (dB) for a system with 5 sensors andactuators. Each block shows the gain from a signal exp(iωt)in sensor k (column) to actuator j (row), ordered upstream todownstream.

The major drawback is that each Newton iteration requiressolving 2r n × n Lyapunov equations until convergence,although recent work simplifies this to 2 equations per iter-ation [27]. Furthermore, the procedure requires an ensembleof random initial conditions to avoid converging to a localminimum. In [16], the optimal placement is computed usingconjugate gradient optimization for the same spatial discretiza-tion n = 100, which becomes computationally expensive asthe grid resolution increases. In this case, gradient descentis more costly than balancing the fully actuated and observedsystem, which comes at a one-time cost of solving 2 Lyapunovequations for the gramians, and 2 Riccati equations for theLQG gain matrices (O(n3) each). Therefore, our algorithmis sensible when the grid discretization is sufficiently fine.Furthermore, our solution is a good starting point for theconvergence of the gradient descent scheme, thus eliminatingthe need for optimization over a large ensemble of randomizedstarting points. QR pivoting runtime scales as O(nr2) and thedeviation of the resulting placement from the H2 optimum(fig. 5) decreases with increasing r.

Figure 5 plots sensor and actuator configurations from theQR algorithm and H2 gradient optimization, which are com-pared with the H2 optimal placements in [16]. The resultingplacements for the cases r = 1 to r = 5 sensors and actuatorsare plotted vertically, and the horizontal axis is the spatialdomain ξ ∈ [−12, 12] with a shaded wave amplification regionin which fluid perturbations are amplified. For each value of r,we apply QR pivoting to the rank r truncated balanced modes.QR pivoting collocates sensors and actuators, indicating thatA is approximately symmetric and hence the direct and adjointmodes (pictured in Fig. 1) are identical up to a scaling factor.In practice, sensors are often slightly downstream to accountfor time delays, so we enforce via the pivoting procedure thatsensors are not placed at previously chosen actuators. The H2

norms of the resulting placement on the y-axis indicate thatthe QR selections closely approximate the optimal placements.The H2 optimal placement [16, Fig. 4] of five sensors andactuators, with H2 norm 27.4, agrees exactly with the H2

optimum and is closely approximated by the QR pivotedplacement (27.8).

Figure 6 compares controller performance between QRpivoting and the H2 optimum via the LQG gain of a givensignal from each sensor to each actuator. The LQG gains

8

are identical to those produced by the H2 optimal methodof Chen and Rowley [16, Fig. 5]. The diffusive nature of thedynamics favors nearly collocating the sensors and actuators,since the high-frequency oscillations mostly propagate to thenearest actuator. This confirms that our framework is useful foroptimizing sensors and actuators. Balanced truncation appliedto the closed loop system is critical to achieving this, sincethe open loop dynamics are unstable and it is shown in [16]that the dominant eigenmodes of the dynamics lead to vastlysuboptimal placements.

VII. DISCUSSION AND OUTLOOK

In this work we develop scalable sensor and actuatorselection whose runtime scales linearly with the number ofstate variables, after a one-time offline computation of thebalanced modes. Our approach relies on balanced modelreduction [21], [23], [24], which hierarchically orders modesby their observability and controllability. We extend EIMs tointerpolate the low-rank balancing modes of the system anddetermine maximally observable and controllable locations(sensor & actuators) in state space. The performance of thisalgorithm is demonstrated on random state-space systems,and optimal H2 control of the linearized Ginzburg-Landaumodel. Our optimized placements vastly exceed the perfor-mance of random placements, and closely approximate H2

optimal placements computed by costly gradient minimizationschemes, but achieved at a fraction of the runtime.

Sensors and actuators are critical for feedback control oflarge high-dimensional complex systems. This work advocatessensor and actuator selection using QR pivots of the directand adjoint modes of a system’s balancing transformation. Theresulting placement is empirically shown to preserve the dy-namics of the full system. The method has deep connections tosystem observability, controllability, modal sampling methodsand classical experimental design criteria. Furthermore, QRpivoting is more computationally efficient than leading greedyand convex optimization methods, and thus critically enlargesthe search space of possible selections. This is particularlyvaluable in spatiotemporal models where high-resolution gridsgenerate a large number of states, and balanced modes and QRmethod exploit the spatial structures.

This work opens a variety of future directions in pivotingsensor and actuator optimization. Rapid advances in datacollection yield extremely large search spaces, for which thecomputation of balanced modes and QR pivoting may beaccelerated using randomized linear algebra. Our method relieson a known model of the dynamics, but it would also beinteresting to generalize the method to data-driven systemidentification models. In addition, point sensors and actua-tors are simplifications of constrained or nonlinear sensingand actuation that may occur in practice. Nonlinear sensingconstraints remain an open challenge.

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